Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1973 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,20,21)(19,23,24,22);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(17,19,20,24)(18,22,21,23);(10,14)(13,15)(18,21)(19,23)(22,24);(2,3)(6,7)(11,16)(13,15)(19,24)(22,23);(1,4,5,8)(2,6,3,7)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239614, #72239653, #72251213, … |
| maximal normal subgroup of | #72239614, #72239653, #72251213, … |
| maximal quotient of | #72239717, #72239726, #72265379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419684);