Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2428 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1988 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)(12,14)(13,16)(18,21)(19,22)(23,24);(9,11,13,12)(10,14,16,15)(18,21)(19,22)(23,24);(1,2,3,4)(9,11,13,12)(10,15,16,14)(17,18,22,24,20,23,19,21);(6,8)(9,12,13,11)(10,14,16,15)(18,23)(21,24)Polycyclic presentation:
2^9|b^2=e,c^2=efi,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72305684, #72305686, #72305708, … |
| maximal normal subgroup of | #72305684, #72305686, #72305708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305770);