Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133093 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#125 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
876 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,10,12,7)(4,8)(5,9)(6,11,15,13)(14,16)(17,18,21,26,22,25,27,30)(19,23,29,31,28,32,20,24);(1,3)(2,6)(4,13)(5,12)(7,8)(9,15)(10,16)(11,14)(17,19)(18,23,25,32)(20,27)(21,29)(22,28)(24,30,31,26);(1,3)(2,7)(4,11)(5,12)(6,8)(9,10)(13,14)(15,16)(17,20,22,29)(18,24,25,31)(19,27,28,21)(23,30,32,26)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,f^b=fghi,f^c=fhi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72293034 |
| maximal normal subgroup of | #72293034 |
| maximal quotient of | #72293138 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40338);