Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514060 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36103 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(18,21)(22,23);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,18,19,21)(20,22,24,23);(10,14)(11,15);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,22)(20,24)(21,23);(2,7)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72284973, #72305296, #72305316 |
| maximal normal subgroup of | #72284973, #72305296, #72305316 |
| maximal quotient of | #72305997, #72306011, #72315954 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7432469);