Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #160780 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,14,13,16,11,15)(17,18,20,24,21,22,19,23);(1,3)(2,5)(4,8)(6,7)(10,14)(11,12)(15,16);(18,22)(23,24);(2,6)(5,7)(17,19,21,20)(18,23,22,24)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=chi,d^a=dg,e^a=eg,e^b=eh,f^b=fgh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257796, #72257797, #72257805, … |
| maximal normal subgroup of | #72257796, #72257797, #72257805, … |
| maximal quotient of | #72264883, #72265302 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 68240);