Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1314 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)(11,14)(12,15)(13,16)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,37)(30,38)(31,43)(32,39)(33,40)(34,41)(35,42)(36,44)(45,47)(46,48);(9,11,14,10)(12,15,13,16)(17,19,23,32)(18,24,27,39)(20,33,35,45)(21,30,22,31)(25,40,42,47)(26,38,28,43)(29,36,46,34)(37,44,48,41);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(17,20,23,35)(18,25,27,42)(19,29,32,46)(21,34,22,36)(24,37,39,48)(26,41,28,44)(30,45,31,33)(38,47,43,40);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(17,21,23,22)(18,26,27,28)(19,30,32,31)(20,34,35,36)(24,38,39,43)(25,41,42,44)(29,45,46,33)(37,47,48,40);(18,27)(19,31)(21,22)(24,38)(25,42)(29,33)(30,32)(34,36)(37,47)(39,43)(40,48)(45,46)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6609294);