Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #94966 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
1378 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15,16,13)(12,14)(17,18,19,22,20,21,23,24);(2,6,8,7)(9,11)(10,13)(12,16)(14,15);(1,3,5,4)(2,7,8,6)(18,21)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gh,d^2=gh,e^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,e^b=eh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72227430, #72227431, #72227432, … |
| maximal normal subgroup of | #72227430, #72227431, #72227432, … |
| maximal quotient of | C2×#94969 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2165);