Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #25775 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,26)(21,27)(22,36)(23,28)(25,33)(29,38)(30,39)(31,46)(32,40)(34,41)(35,42)(37,43)(44,47)(45,48);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,23,32)(18,24,28,40)(20,29,34,44)(21,30,22,31)(25,37,42,48)(26,38,41,47)(27,39,36,46)(33,43,35,45);(11,16)(13,15)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,36)(27,41)(28,42)(30,43)(31,45)(32,44)(38,46)(39,47)(40,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20)(18,26)(19,29)(21,35)(22,33)(23,34)(24,38)(25,36)(27,42)(28,41)(30,45)(31,43)(32,44)(37,46)(39,48)(40,47);(2,5)(6,8)(17,21,23,22)(18,27,28,36)(19,30,32,31)(20,33,34,35)(24,39,40,46)(25,41,42,26)(29,43,44,45)(37,47,48,38)Polycyclic presentation:
2^9|b^2=fi,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6678825);