Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#102 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
398 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,9,12)(7,13,14,20)(8,16,17,11)(10,18,19,15)(22,28)(23,32)(24,25)(26,34)(29,35)(30,36)(31,33);(5,7)(6,10,12,19)(8,15)(9,14)(11,13,16,20)(17,18)(21,22,25,29,26,30,23,33,27,31,32,36,34,35,24,28);(1,3,4,2)(7,14)(10,19)(13,20)(15,18)(21,23,34,25,27,24,26,32)(22,28,35,36,31,33,30,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=egi,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fgi,g^a=gi,h^a=hi,c^b=cgh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35809);