Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,21,23)(19,27,28,41)(20,31,32,25)(22,35,34,26)(24,37,36,30)(29,43,42,40)(33,39,47,46)(38,44,45,48);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(25,31)(26,35)(29,42)(30,37)(39,46)(44,48);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,30,21,28,34,37)(18,24,26,27,23,36,35,41)(20,29,33,44,32,42,47,48)(25,38,39,40,31,45,46,43);(2,6)(5,7)(11,16)(13,15)(17,20)(18,25)(19,29)(21,32)(22,33)(23,31)(24,38)(26,39)(27,40)(28,42)(30,44)(34,47)(35,46)(36,45)(37,48)(41,43);(10,14)(13,15)(19,28)(20,32)(24,36)(25,31)(27,41)(30,37)(33,47)(39,46)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6816205);