Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142969 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #3490 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(2,6)(4,11)(7,14)(8,12)(9,16)(13,15)(17,18,21,24)(19,22,27,32)(20,28,29,23)(25,30,26,31);(1,2)(3,7)(4,12)(5,8)(6,11)(9,13)(10,15)(14,16)(17,19)(18,22)(20,25)(21,27)(23,30)(24,32)(26,29)(28,31);(1,3,5,10)(2,7,8,15)(4,9,11,16)(6,13,12,14)(19,25)(20,29)(22,30)(23,28)(26,27)(31,32)Polycyclic presentation:
2^9|a^2=gi,c^2=g,d^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,h^b=hi,d^c=dh,f^c=fgi,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72303935, #72304001, #72304010 |
| maximal normal subgroup of | #72303935, #72304001, #72304010 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50194);