Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #92889 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1272 |
| commutator | C2×C42 | C2×C8 |
| Frattini, cosocle | C4×#95 | C22 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C4×#95 ▷ C23×C4 ▷ C22 ▷ C1 | |
340 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 48 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 — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,18,20,14,19,23,21,24)(12,15)(13,17)(16,22);(2,5)(6,7)(9,11,13,19)(10,14)(12,21,22,20)(15,18)(16,23)(17,24)Polycyclic presentation:
2^9|a^2=d,b^2=f,c^2=fi,d^2=g,e^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,c^b=cf,d^b=def,e^b=ei,g^b=gi,d^c=dhi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227432 |
| maximal normal subgroup of | #72227432 |
| maximal quotient of | #72268638 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 85);