Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,19,22,20,21,23,24);(2,6)(3,4)(11,16)(13,15);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15);(1,4,5,3)(2,7,6,8)(10,14)(13,15);(18,21)(22,24)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=cfg,d^b=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264624, #72264630, #72264635, … |
| maximal normal subgroup of | #72264624, #72264630, #72264635, … |
| maximal quotient of | C2×#6407637 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314833);