Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480656 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #2403 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10,13,16,12,15,20,23)(11,17,19,24,18,14,21,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,21)(23,24);(2,5)(6,8)(10,15)(11,18)(19,21)(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,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,g^c=gh,g^f=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72236776, #72240280, #72312481, … |
| maximal normal subgroup of | #72236776, #72240280, #72312481, … |
| maximal quotient of | #72317673 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403714);