Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61262 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,30)(20,33)(21,28)(22,27)(23,39)(24,29)(25,31)(26,36)(32,44)(34,48)(35,43)(37,46)(38,47)(40,45)(41,42);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,22,31)(18,25,27,30)(20,34,35,41)(21,36,38,37)(23,32,24,40)(26,28,46,47)(29,44,39,45)(33,42,43,48);(17,20,22,35)(18,26,27,46)(19,21,31,38)(23,37,24,36)(25,42,30,48)(28,45,47,44)(29,43,39,33)(32,41,40,34);(2,7)(6,9)(8,15)(14,16)(18,27)(25,30)(26,46)(28,47)(29,39)(33,43)(42,48)(44,45);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21,22,38)(18,28,27,47)(19,20,31,35)(23,34,24,41)(25,43,30,33)(26,45,46,44)(29,42,39,48)(32,36,40,37)Polycyclic presentation:
2^9|b^2=f,c^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580778);