Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,22,19,21,23,24);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(18,21)(22,24);(10,14)(13,16);(3,8)(5,7);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=h,b^2=g,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=dfi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72309342, #72309348, #72309349, … |
| maximal normal subgroup of | #72309342, #72309348, #72309349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293708);