Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132559 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #52395 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
742 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 742 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)(3,11,12,8)(4,7)(5,15,13,9)(6,10)(14,16)(17,18,21,25,22,26,20,24)(19,23,29,30,27,32,28,31);(1,3)(2,7)(4,13)(5,14)(6,12)(8,9)(10,16)(11,15)(17,19)(18,23)(20,29)(21,28)(22,27)(24,31)(25,30)(26,32);(1,4,6,14)(2,8)(3,5,12,13)(7,15)(9,16)(10,11)(18,24,26,25)(19,27)(20,21)(23,30,32,31)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72227761 |
| maximal normal subgroup of | #72227761 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40238);