Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18,21,25)(19,27,30,23)(20,31,33,24)(22,26,35,42)(28,38,46,45)(29,43,36,39)(32,40,47,48)(34,44,37,41);(1,2)(3,7)(4,8)(5,6)(17,19)(18,23)(20,32)(21,30)(22,36)(24,40)(25,27)(26,43)(28,34)(29,35)(31,48)(33,47)(37,46)(38,41)(39,42)(44,45);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,21,33)(18,24,25,31)(19,28,30,46)(22,37,35,34)(23,38,27,45)(26,44,42,41)(29,47,36,32)(39,48,43,40);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(18,25)(20,33)(23,27)(26,42)(28,46)(32,47)(34,37)(39,43);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(19,29)(22,35)(23,39)(26,42)(27,43)(28,47)(30,36)(32,46)(34,37)(38,48)(40,45)(41,44)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=gi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6646642);