Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | C2×C22⋊C8 | C23 |
| socle | C22 | #1459 |
| prime radical | C2×#120 | C8 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
389 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12,13,7)(5,8)(6,11,14,16)(10,15)(17,18,20,23,21,24,29,30,22,25,19,26,27,31,28,32);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16);(3,10)(6,14)(9,15)(11,16)(17,19,21,28,22,20,27,29)(18,23,24,30,25,26,31,32)Polycyclic presentation:
2^9|a^2=e,c^2=g,e^2=g,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=ch,e^b=ef,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29560);