Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #127185 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C24 | #95 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
539 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,8,6,10,5,9)(3,11,12,7,14,15,13,16)(17,18,21,25,22,26,34,42)(19,27,30,44,28,43,46,48)(20,24,31,38,32,39,33,41)(23,35,37,29,36,47,40,45);(1,3,5,13,6,14,4,12)(2,7,8,15,10,16,9,11)(17,19)(18,23,26,36)(20,29)(21,30)(22,28)(24,27,39,43)(25,40,42,37)(31,47)(32,45)(33,35)(34,46)(38,48,41,44);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(19,28)(23,36)(27,43)(29,45)(30,46)(35,47)(37,40)(44,48)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ghi,d^2=gh,e^2=gh,f^2=ghi,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=ci,e^b=eh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34662);