Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
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 | |
5622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,8)(9,10,11,12)(14,15)(17,18,21,19)(20,23,24,22);(2,4)(6,8)(13,14,16,15)(18,19)(22,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,16,14)(17,19,21,18)(20,23,24,22);(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=i,b^2=eh,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72329917, #72330048, #72330255, … |
| maximal normal subgroup of | #72329917, #72330048, #72330255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 61328);