Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C8 | C23 |
| socle | C22 | C2×#327 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
380 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,8)(5,14,15,9)(6,10)(7,12,16,13)(18,22)(19,25)(20,27)(23,32)(24,31)(26,29)(28,30);(1,3,6,4)(2,7,10,16)(5,12,15,13)(8,9,11,14)(17,18,21,24)(19,26,28,32)(20,29,27,23)(22,30,31,25);(1,4,6,3)(2,8,10,11)(5,13,15,12)(7,9,16,14)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=hi,d^2=gi,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29739);