Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,27,29,39)(20,31,32,24)(21,34)(22,26)(23,28)(25,35)(30,44,43,40)(33,38,46,45)(36,48)(37,42)(41,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,30,22,29,35,43)(18,23,25,36,26,37,34,47)(20,28,33,41,32,42,46,48)(24,27,38,40,31,39,45,44);(3,9)(6,12)(10,15)(13,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=h,f^2=i,b^a=bd,d^a=dg,f^a=fi,g^a=gi,c^b=ch,e^b=egi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34207);