Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C24 |
| commutator | C24 | C23×C4 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
4422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(6,8)(10,12)(13,14,16,15)(17,18,20,21)(19,22,23,24);(1,2,3,4)(5,6,7,8)(18,21)(22,24);(5,6)(7,8)(9,10,11,12)(13,15,16,14)(17,19,20,23)(18,22,21,24);(1,2)(3,4)(5,6)(7,8)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=egh,c^2=fgi,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de116 conjugacy classes — view the full character table.
| maximal subgroup of | #72267316, #72267318, #72267320, … |
| maximal normal subgroup of | #72267316, #72267318, #72267320, … |
| maximal quotient of | #72238841, #72238845, C2×#155376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 62572);