Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×C42 | C2×C8 |
| Frattini, cosocle | #1842 | C22 |
| socle | C22 | #1325 |
| prime radical | C23 | #324 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1842 ◁ G | |
| LECS | G ▷ #1842 ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2)(3,10,12,6)(4,8,11,7)(5,9)(13,14,16,15)(17,18,20,25,21,26,31,39,22,24,30,36,32,37,43,46)(19,23,28,33,29,35,42,44,27,34,40,38,41,45,48,47);(1,3,11,16,5,13,4,12)(2,6,8,14,9,15,7,10)(17,19,22,27)(18,23)(20,28,30,40)(21,29,32,41)(24,34)(25,38)(26,35)(31,42,43,48)(33,36)(37,45)(39,47)(44,46)Polycyclic presentation:
2^9|a^2=d,b^2=eg,c^2=ghi,d^2=f,e^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ch,d^b=degh,d^c=dgi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 934);