Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186967 |
| 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×#3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3444 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)(10,14,16,19)(15,23,24,22);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,19)(13,16)(14,21)(17,22)(18,24)(20,23);(1,3)(2,5)(4,8)(6,7)(9,11,13,18)(10,15,16,24)(12,20,21,17)(14,22,19,23);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239056 |
| maximal normal subgroup of | #72239056 |
| maximal quotient of | #72239145 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94166);