Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632068 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,9)(3,10,12,7)(4,8,5,13)(11,16,14,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,34,37,42,38,43,50,40)(35,39,46,54,45,53,60,51)(36,41,49,58,48,57,62,55)(44,52,59,64,47,56,61,63);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,22)(20,27)(21,26)(23,31)(24,30)(25,28)(29,32)(33,35)(34,39)(36,47)(37,46)(38,45)(40,51)(41,56)(42,54)(43,53)(44,48)(49,61)(50,60)(52,57)(55,64)(58,63)(59,62);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,20,21,28)(18,23,24,32)(19,25,26,27)(22,29,30,31)(34,40)(37,50)(39,51)(41,55)(42,43)(46,60)(49,62)(52,63)(53,54)(56,64)(57,58)(59,61);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(19,26)(22,30)(25,27)(29,31)(33,36)(34,41)(35,44)(37,49)(38,48)(39,52)(40,55)(42,58)(43,57)(45,47)(46,59)(50,62)(51,63)(53,56)(54,64)(60,61);(2,9)(7,10)(8,13)(15,16)(18,24)(22,30)(23,32)(29,31)(35,45)(36,48)(39,53)(41,57)(46,60)(49,62)(51,54)(55,58)Polycyclic presentation:
2^9|a^2=fg,c^2=fh,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6550476);