Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,16)(11,18,21,22)(12,20,17,23)(19,24);(1,3,5,2)(4,7,8,6)(9,10,13,16)(11,20,21,23)(12,22,17,18)(14,19,15,24);(1,4,5,8)(2,6,3,7)(9,11,15,18,13,21,14,22)(10,17,24,20,16,12,19,23);(2,3)(6,7)(9,12)(10,18)(11,19)(13,17)(14,20)(15,23)(16,22)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=efh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296425);