Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1758 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,25,21,27,22,28)(19,29,35,39,31,24,32,37)(20,26,36,40,33,41,34,42)(30,43,46,48,44,38,45,47);(2,5)(3,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,30)(21,31)(22,32)(23,35)(25,37)(26,38)(27,29)(28,39)(33,44)(34,45)(36,46)(40,47)(41,43)(42,48);(11,16)(13,15)(18,25)(22,23)(24,37)(26,40)(27,28)(29,39)(32,35)(34,36)(38,47)(41,42)(43,48)(45,46);(2,5)(6,8)(10,14)(13,15)(17,20,21,33)(18,26,27,41)(19,30,31,44)(22,34,23,36)(24,38,29,43)(25,40,28,42)(32,45,35,46)(37,47,39,48);(1,3,4,7)(2,6,5,8)(19,31)(20,33)(24,29)(26,41)(32,35)(34,36)(37,39)(40,42)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6709855);