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×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3496 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)(10,15,17,21)(16,22,24,23);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,24)(20,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,18)(14,19)(15,22)(17,24)(21,23);(9,12,14,13)(10,15,17,21)(11,18,19,20)(16,23,24,22)Polycyclic presentation:
2^9|a^2=g,d^2=i,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 | #72239065 |
| maximal normal subgroup of | #72239065 |
| maximal quotient of | #72239139 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92705);