Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #513316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×D32 |
| commutator, Frattini, cosocle | C32 | C24 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ #504 ◁ #2258 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 282 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:
(1,2,8,15)(3,10,16,39)(4,9,25,38)(5,26,29,79)(6,30,32,82)(7,33,31,14)(11,44,47,21)(12,48,50,22)(13,51,53,27)(17,34,55,84)(18,63,56,108)(19,66,57,110)(20,67,58,43)(23,52,75,105)(24,54,74,37)(28,80,71,78)(35,88,90,69)(36,91,92,68)(40,96,93,59)(41,99,94,60)(42,101,95,64)(45,104,89,73)(46,70,83,76)(49,72,77,81)(61,102,106,127)(62,103,107,87)(65,119,109,118)(85,124,122,112)(86,126,123,111)(97,128,125,115)(98,113,121,116)(100,114,117,120);(2,9)(4,21)(5,22)(6,27)(7,31)(10,34)(11,12)(13,52)(14,54)(15,38)(17,59)(18,60)(19,64)(20,58)(23,73)(24,76)(25,44)(26,35)(28,81)(29,48)(30,36)(32,51)(33,37)(39,84)(40,41)(42,102)(43,103)(45,78)(46,77)(47,50)(49,83)(53,105)(55,96)(56,99)(57,101)(61,115)(62,116)(63,85)(65,120)(66,86)(67,87)(68,69)(70,74)(71,72)(75,104)(79,90)(80,89)(82,92)(88,91)(93,94)(95,127)(97,118)(98,117)(100,121)(106,128)(107,113)(108,122)(109,114)(110,123)(111,112)(119,125)(124,126);(1,3)(2,10)(4,17)(5,18)(6,19)(7,20)(8,16)(9,34)(11,40)(12,41)(13,42)(14,43)(15,39)(21,59)(22,60)(23,61)(24,62)(25,55)(26,63)(27,64)(28,65)(29,56)(30,66)(31,58)(32,57)(33,67)(35,85)(36,86)(37,87)(38,84)(44,96)(45,97)(46,98)(47,93)(48,99)(49,100)(50,94)(51,101)(52,102)(53,95)(54,103)(68,111)(69,112)(70,113)(71,109)(72,114)(73,115)(74,107)(75,106)(76,116)(77,117)(78,118)(79,108)(80,119)(81,120)(82,110)(83,121)(88,124)(89,125)(90,122)(91,126)(92,123)(104,128)(105,127);(3,16)(10,39)(17,55)(18,56)(19,57)(20,58)(34,84)(40,93)(41,94)(42,95)(43,67)(59,96)(60,99)(61,106)(62,107)(63,108)(64,101)(65,109)(66,110)(85,122)(86,123)(87,103)(97,125)(98,121)(100,117)(102,127)(111,126)(112,124)(113,116)(114,120)(115,128)(118,119)Polycyclic presentation:
2^9|a^2=i,e^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=be,e^a=ef,f^a=fg,g^a=gh,h^a=hi,e^b=ef,f^b=fg,g^b=gh,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321377 |
| maximal normal subgroup of | #72321377 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420513);