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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1360 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)(9,10,13,12)(11,14,16,15)(17,18,23,28)(19,24,31,42)(20,32,37,47)(21,27,22,39)(25,35,44,36)(26,38,41,30)(29,43,34,45)(33,40,48,46);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,33)(21,30)(22,38)(23,31)(25,43)(26,27)(28,42)(29,36)(32,40)(34,35)(37,48)(39,41)(44,45)(46,47);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,22,36,23,37,21,35)(18,25,27,32,28,44,39,47)(19,29,30,33,31,34,38,48)(24,40,41,43,42,46,26,45);(1,4,5,3)(2,6,8,7)(11,16)(14,15)(17,19,21,38,23,31,22,30)(18,26,39,42,28,41,27,24)(20,34,35,33,37,29,36,48)(25,43,47,40,44,45,32,46);(2,8)(6,7)(17,21,23,22)(18,27,28,39)(19,30,31,38)(20,35,37,36)(24,26,42,41)(25,32,44,47)(29,33,34,48)(40,45,46,43)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fh,d^2=fgi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7510904);