Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #13742 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D32 |
| commutator, Frattini, cosocle | C32 | C24 |
| socle | C2 | C22×D32 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
284 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2)(3,16)(4,12)(5,25)(6,32)(7,37)(8,40)(9,11)(10,18)(13,28)(14,27)(15,34)(17,57)(19,64)(20,66)(21,56)(22,68)(23,73)(24,49)(26,54)(29,87)(30,90)(31,75)(33,52)(35,96)(36,92)(38,58)(39,98)(41,59)(42,63)(43,60)(44,79)(45,78)(46,101)(47,86)(48,83)(50,71)(51,70)(53,113)(55,62)(61,118)(65,120)(67,117)(69,111)(72,122)(74,110)(76,81)(77,84)(80,109)(82,93)(85,100)(88,104)(89,127)(91,115)(94,114)(95,107)(97,112)(99,116)(102,123)(103,125)(105,119)(106,128)(108,121)(124,126);(1,3)(2,10)(4,17)(5,26)(6,33)(7,38)(8,41)(9,21)(11,42)(12,43)(13,52)(14,54)(15,55)(16,25)(18,28)(19,27)(20,34)(22,69)(23,74)(24,67)(29,88)(30,91)(31,80)(32,44)(35,97)(36,95)(37,45)(39,99)(40,46)(47,107)(48,109)(49,105)(50,110)(51,111)(53,114)(56,75)(57,68)(58,87)(59,90)(60,71)(61,70)(62,113)(63,86)(64,76)(65,83)(66,77)(72,123)(73,104)(78,81)(79,84)(82,94)(85,101)(89,128)(92,102)(93,115)(96,106)(98,103)(100,112)(108,118)(116,127)(117,122)(119,121)(120,124)(125,126);(2,11)(10,42)(12,40)(13,47)(14,48)(15,49)(16,56)(25,75)(32,92)(37,98)(43,46)(44,102)(45,103)(50,100)(51,93)(52,107)(53,108)(54,109)(55,105)(57,59)(58,116)(64,120)(66,117)(68,90)(73,96)(76,124)(77,122)(87,127)(104,106)(110,112)(111,115)(114,118);(1,4,9,8)(2,12,11,40)(3,17,21,41)(5,22,31,30)(6,23,36,35)(7,24,39,34)(10,43,42,46)(13,50,47,100)(14,51,48,93)(15,37,49,98)(16,57,56,59)(18,60,63,101)(19,61,65,94)(20,38,67,99)(25,68,75,90)(26,69,80,91)(27,70,83,82)(28,71,86,85)(29,72,89,84)(32,73,92,96)(33,74,95,97)(44,104,102,106)(45,105,103,55)(52,110,107,112)(53,76,108,124)(54,111,109,115)(58,117,116,66)(62,78,119,125)(64,118,120,114)(77,87,122,127)(79,88,123,128)(81,121,126,113)Polycyclic presentation:
2^9|d^2=i,e^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=ci,e^a=ef,f^a=fg,g^a=gh,h^a=hi,e^b=ef,f^b=fg,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321113, #72321197, #72321203, … |
| maximal normal subgroup of | #72321113, #72321197, #72321203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420511);