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 | #45136 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1562 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)(19,24)(20,31)(21,37)(22,39)(23,28)(25,36)(26,30)(27,33)(29,45)(32,43)(34,40)(35,47)(38,41)(42,46)(44,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,28,26)(20,32,35,34)(21,22,38,33)(25,42,44,45)(27,37,39,41)(29,36,46,48)(31,43,47,40);(10,14)(11,16)(17,20,33,48,23,35,22,36)(18,25,39,47,28,44,27,31)(19,29,38,32,30,46,21,34)(24,40,37,42,26,43,41,45);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,30,23,38,33,19)(18,26,27,37,28,24,39,41)(20,34,36,46,35,32,48,29)(25,43,31,42,44,40,47,45);(2,5)(6,8)(10,14)(13,15)(20,35)(25,44)(29,46)(31,47)(32,34)(36,48)(40,43)(42,45)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=fg,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,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7502805);