Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133828 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1242 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1242 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,18,20,14)(12,17,21,15)(13,16)(19,22)(23,24);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,19)(13,20)(15,22)(16,18)(17,23)(21,24);(10,15)(12,21)(14,22)(16,17)(18,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=gi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=ei,f^c=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72292879, C2×#133855 |
| maximal normal subgroup of | #72292879, C2×#133855 |
| maximal quotient of | C2×#133855, #72293078, #72293079 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41051);