Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133089 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
876 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 78 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)(6,10,14,16)(11,15)(17,18)(19,27,29,32)(20,25,21,24)(22,26)(23,28)(30,31);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19,22,29)(18,23,26,31)(20,28,21,30)(24,27,25,32);(4,13)(7,12)(9,15)(10,16)(17,20)(18,24)(19,28)(21,22)(23,27)(25,26)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fh,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72227670, #72227681, #72227683 |
| maximal normal subgroup of | #72227670, #72227681, #72227683 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40414);