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 | |
1233 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)(11,17)(12,16)(13,21,20,15)(14,18)(19,23,24,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22);(2,5)(6,8)(10,15)(11,18)(14,22)(16,21)(17,23)(19,24);(1,3,4,7)(2,6,5,8)(10,16)(14,17)(15,21)(22,23)Polycyclic presentation:
2^9|a^2=g,d^2=ei,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, #72240267, #72263649, … |
| maximal normal subgroup of | #72236776, #72240267, #72263649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403702);