Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(10,13)(14,16)(17,18,21,24)(19,22,20,23);(1,2)(3,4)(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,21,20)(18,22,24,23);(2,4)(6,8)(9,11,12,15)(10,14,13,16);(6,8)(10,13)(14,16)(18,23)(19,20)(22,24)Polycyclic presentation:
2^9|a^2=fi,b^2=efi,c^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239669, #72239681, #72239711, … |
| maximal normal subgroup of | #72239669, #72239681, #72239711, … |
| maximal quotient of | #72250141, #72322070 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 333914);