Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7540 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1415 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C2 | C2×#1415 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 128 points:
(1,2,4,10,8,15,26,14)(3,16,19,64,22,67,17,62)(5,27,23,81,32,61,85,92)(6,33,24,86,36,51,88,101)(7,37,25,13,39,48,89,60)(9,40,42,18,45,72,41,75)(11,49,46,70,54,105,53,30)(12,55,47,29,58,83,57,35)(20,77,76,119,78,117,68,124)(21,79,38,66,80,116,69,123)(28,56,93,121,73,63,82,113)(31,98,84,91,100,122,125,118)(34,65,103,111,71,50,87,115)(43,99,107,102,109,128,106,95)(44,74,108,104,110,127,59,94)(52,96,112,126,120,97,90,114);(1,3)(2,9)(4,17)(5,28)(6,34)(7,38)(8,22)(10,41)(11,50)(12,56)(13,59)(14,42)(15,45)(16,61)(18,70)(19,26)(20,35)(21,25)(23,82)(24,87)(27,67)(29,78)(30,75)(31,99)(32,73)(33,43)(36,71)(37,44)(39,69)(40,105)(46,111)(47,113)(48,110)(49,72)(51,109)(52,119)(53,115)(54,65)(55,68)(57,121)(58,63)(60,108)(62,92)(64,81)(66,122)(74,126)(76,83)(77,112)(79,118)(80,89)(84,95)(85,93)(86,106)(88,103)(90,117)(91,116)(94,97)(96,104)(98,123)(100,128)(101,107)(102,125)(114,127)(120,124);(1,4,8,26)(2,10,15,14)(3,17,22,19)(5,23,32,85)(6,24,36,88)(7,25,39,89)(9,41,45,42)(11,46,54,53)(12,47,58,57)(13,48,60,37)(16,62,67,64)(18,40,75,72)(20,68,78,76)(21,69,80,38)(27,81,61,92)(28,82,73,93)(29,83,35,55)(30,49,70,105)(31,84,100,125)(33,86,51,101)(34,87,71,103)(43,106,109,107)(44,59,110,108)(50,111,65,115)(52,112,120,90)(56,113,63,121)(66,79,123,116)(74,94,127,104)(77,124,117,119)(91,122,118,98)(95,128,102,99)(96,126,97,114)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=fgi,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,f^b=fg,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321113, #72321178, #72321207, … |
| maximal normal subgroup of | #72321113, #72321178, #72321207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60761);