Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130102 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #45599 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
740 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 740 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) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10)(11,17,18,24)(12,20,22,15)(13,16)(14,19)(21,23);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22);(1,4,6,5)(2,7,3,8)(11,18)(14,23)(15,20)(19,21)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72338680, #72338682, #72338689, … |
| maximal normal subgroup of | #72338680, #72338682, #72338689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37551);