Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8516 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 112 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)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,20,25,22,26,33,40,21,24,32,37,34,38,45,47)(19,27,29,39,30,41,43,48,28,23,31,35,42,36,44,46);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19)(18,23)(20,31)(21,28)(22,30)(24,27)(25,39)(26,36)(29,32)(33,44)(34,42)(35,37)(38,41)(40,48)(43,45)(46,47);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)(19,28)(23,27)(29,31)(30,42)(35,39)(36,41)(43,44)(46,48)Polycyclic presentation:
2^9|a^2=f,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=ci,d^b=dgi,f^b=fi,f^c=fgi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36880);