Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,23,21,24,20,19);(1,2,3,4)(10,13)(14,15)(17,19,20,24,21,23,22,18);(9,11,12,16)(10,14,13,15)(17,20,21,22)(18,23,24,19);(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dg,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72237122, #72237141, #72237146, … |
| maximal normal subgroup of | #72237122, #72237141, #72237146, … |
| maximal quotient of | #72239244, #72239286, #72239307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300494);