Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480670 |
| 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 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1091 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)(9,10,14,18,13,15,21,22)(11,17,19,24,12,16,20,23);(1,2)(3,6)(4,5)(7,8)(10,15)(11,12)(18,22)(19,20);(2,5)(6,8)(9,11,13,12)(10,16,15,17)(14,19,21,20)(18,24,22,23);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,17,15,16)(14,20,21,19)(18,24,22,23)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,g^c=gh,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72318151, #72318164, #72318190, … |
| maximal normal subgroup of | #72318151, #72318164, #72318190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403758);