Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116001 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17,19,23)(12,15)(13,16)(14,20,18,21)(22,24);(9,10)(11,18,19,14)(12,15)(13,16)(17,20,23,21)(22,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,21)(13,19)(15,23)(16,18)(17,24)(20,22)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eh,g^c=ghi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234285, #72258988, #72259001, … |
| maximal normal subgroup of | #72234285, #72258988, #72259001, … |
| maximal quotient of | #72234302, C2×#116001 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23197);