Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(10,16)(11,18)(12,21)(13,14)(20,22)(23,24);(1,2)(3,4)(9,10)(11,19)(12,15)(13,21)(14,16)(17,20)(18,23)(22,24);(2,4)(9,11,15,20)(10,17,12,19)(13,18,14,22)(16,23,21,24);(9,12)(10,15)(11,19)(13,16)(14,21)(17,20)(18,23)(22,24)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72316753 |
| maximal normal subgroup of | #72316753 |
| maximal quotient of | #72317090 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266858);