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 | #45316 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1559 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)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,31,41)(20,32,36,25)(21,26,22,27)(29,43,34,39)(30,40,37,45)(33,48,47,42)(35,46,38,44);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,33)(21,30)(22,37)(23,31)(25,42)(26,40)(27,45)(28,41)(29,35)(32,48)(34,38)(36,47)(39,44)(43,46);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19)(18,24)(20,34)(21,37)(22,30)(23,31)(25,43)(26,45)(27,40)(28,41)(29,36)(32,39)(33,38)(35,47)(42,46)(44,48);(10,14)(13,16)(17,20,22,35,23,36,21,38)(18,25,27,44,28,32,26,46)(19,29,30,33,31,34,37,47)(24,39,40,42,41,43,45,48);(1,4,5,3)(2,7,8,6)(17,21,23,22)(18,26,28,27)(19,30,31,37)(20,35,36,38)(24,40,41,45)(25,44,32,46)(29,47,34,33)(39,48,43,42)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7520234);