Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631116 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1147 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)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31)(33,34,38,43,39,44,37,42)(35,40,46,54,47,55,51,57)(36,48,49,56,50,58,52,41)(45,59,62,64,61,63,60,53);(2,6)(7,12)(8,13)(14,16)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29)(33,35)(34,40)(36,45)(37,51)(38,46)(39,47)(41,53)(42,57)(43,54)(44,55)(48,59)(49,62)(50,61)(52,60)(56,64)(58,63);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31)(33,36,39,50)(34,41,44,56)(35,45,47,61)(37,49,38,52)(40,53,55,64)(42,48,43,58)(46,60,51,62)(54,63,57,59);(3,9)(7,12)(10,15)(14,16)(18,24)(20,28)(22,30)(25,27)(33,37,39,38)(34,42,44,43)(35,46,47,51)(36,49,50,52)(40,54,55,57)(41,48,56,58)(45,60,61,62)(53,63,64,59);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(35,47)(40,55)(45,61)(46,51)(53,64)(54,57)(59,63)(60,62)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^b=di,e^b=ei,d^c=dfh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6539144);