Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1129 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,27)(21,24)(23,29)(25,32)(26,31)(28,30)(33,34,37,41,38,42,48,54)(35,39,45,51,44,50,57,61)(36,40,47,53,46,52,58,62)(43,49,56,60,55,59,63,64);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,21,26)(18,22,24,31)(20,28,29,25)(23,32,27,30)(33,35)(34,39)(36,43)(37,45)(38,44)(40,49)(41,51)(42,50)(46,55)(47,56)(48,57)(52,59)(53,60)(54,61)(58,63)(62,64);(3,10)(6,9)(11,16)(13,15)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32)(33,36)(34,40)(35,43)(37,47)(38,46)(39,49)(41,53)(42,52)(44,55)(45,56)(48,58)(50,59)(51,60)(54,62)(57,63)(61,64);(2,7)(3,10)(8,14)(11,16)(18,24)(22,31)(23,27)(30,32)(36,46)(40,52)(43,55)(47,58)(49,59)(53,62)(56,63)(60,64);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(35,44)(39,50)(43,55)(45,57)(49,59)(51,61)(56,63)(60,64)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314913);