Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185503 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3488 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,18,19,15)(12,14)(13,17)(16,21)(20,23,24,22);(1,3)(2,5)(4,8)(6,7)(10,14)(12,21)(15,22)(16,17)(18,23)(20,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,19)(14,22)(16,23)(17,18)(21,24);(11,19)(15,18)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=g,e^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,g^b=ghi,d^c=di,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238943, #72239721 |
| maximal normal subgroup of | #72238943, #72239721 |
| maximal quotient of | #72238964, #72239741 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92699);