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 | #45265 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1557 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)(11,15)(12,14)(13,16)(17,18,24,28)(19,25,31,42)(20,32,37,48)(21,38,30,41)(22,27,23,39)(26,35,45,36)(29,46,47,43)(33,44,34,40);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,33)(21,23)(22,30)(24,31)(26,43)(27,41)(28,42)(29,35)(32,44)(34,37)(36,47)(38,39)(40,48)(45,46);(10,14)(11,16)(17,20,23,36,24,37,22,35)(18,26,27,32,28,45,39,48)(19,29,30,34,31,47,21,33)(25,40,38,46,42,44,41,43);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,31,24,30,23,19)(18,25,39,41,28,42,27,38)(20,34,35,29,37,33,36,47)(26,44,48,46,45,40,32,43);(2,5)(6,8)(10,14)(13,15)(17,22,24,23)(18,27,28,39)(19,30,31,21)(20,35,37,36)(25,41,42,38)(26,32,45,48)(29,34,47,33)(40,43,44,46)Polycyclic presentation:
2^9|a^2=i,c^2=h,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=cgi,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, 7502908);