Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1362 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,33)(21,28)(22,27)(23,31)(25,36)(26,37)(29,38)(30,39)(32,41)(34,42)(35,43)(40,46)(44,48)(45,47);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,34)(21,30)(22,31)(24,27)(25,42)(26,39)(28,41)(29,35)(32,37)(33,47)(36,45)(38,44)(40,43)(46,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,24)(20,29)(21,32)(22,31)(23,27)(25,40)(26,39)(28,41)(30,37)(33,38)(34,35)(36,46)(42,43)(44,47)(45,48);(1,4,5,3)(2,8,6,7)(10,14)(11,16)(17,20,21,35,22,36,37,48)(18,25,28,44,27,33,26,43)(19,29,32,34,31,46,30,45)(23,38,39,42,24,40,41,47);(2,6)(7,8)(10,14)(13,15)(17,21,22,37)(18,26,27,28)(19,30,31,32)(20,35,36,48)(23,39,24,41)(25,43,33,44)(29,45,46,34)(38,42,40,47)Polycyclic presentation:
2^9|b^2=g,c^2=fgi,d^2=fhi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7523378);