Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480454 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35076 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1304 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)(3,6)(4,5)(7,8)(9,10,13,16,12,15,18,23)(11,17,20,24,19,14,21,22);(9,11,18,20)(10,14,16,22)(12,19,13,21)(15,17,23,24);(2,5)(6,8)(10,15)(14,17)(16,23)(22,24);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236583, #72240280, #72278399, … |
| maximal normal subgroup of | #72236583, #72240280, #72278399, … |
| maximal quotient of | #72317654 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402793);