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 | #23390 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1743 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,33)(21,27)(22,24)(23,37)(25,32)(26,35)(29,43)(30,45)(31,38)(34,41)(36,40)(39,47)(42,44)(46,48);(1,3,4,8)(2,5,6,7)(9,11,12,16)(10,13,14,15)(18,24)(19,29)(26,33)(27,37)(30,44)(31,32)(40,41)(46,47);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,19,21,31,22,29,23,32)(18,25,37,43,24,38,27,28)(20,30,36,47,35,44,34,46)(26,39,40,45,33,48,41,42);(2,6)(5,7)(10,14)(13,15)(17,20)(18,26)(19,30)(21,36)(22,35)(23,34)(24,33)(25,39)(27,41)(28,42)(29,44)(31,47)(32,46)(37,40)(38,48)(43,45);(3,8)(5,7)(17,21,22,23)(18,27,24,37)(19,31,29,32)(20,34,35,36)(25,28,38,43)(26,40,33,41)(30,46,44,47)(39,45,48,42)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,e^2=i,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^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6815986);