Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C24 |
| commutator | C23 | C22×C42 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
4050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10,11,12)(17,18,20,21)(19,22,23,24);(5,6,8,7)(14,16)(18,21)(19,23);(9,10)(11,12)(13,14,15,16)(17,18)(19,22)(20,21)(23,24);(1,3,4,2)(5,7,8,6)(9,10)(11,12)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eg,d^2=efh,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de128 conjugacy classes — view the full character table.
| maximal subgroup of | #72267735, #72267736, #72267737, … |
| maximal normal subgroup of | #72267735, #72267736, #72267737, … |
| maximal quotient of | #72299904 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 61212);