Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
856 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)(3,6)(4,7)(5,8)(11,20,22,16)(12,21,15,23);(1,3,6,2)(4,8,5,7)(9,10)(11,21)(12,16)(13,17)(14,19)(15,20)(18,24)(22,23);(1,4,6,5)(2,7,3,8)(9,11,13,16,14,22,24,20)(10,15,17,21,19,12,18,23);(1,5,6,4)(2,7,3,8)(9,12)(10,16)(11,18)(13,23)(14,15)(17,22)(19,20)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=efh,d^2=e,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.
| maximal quotient of | #72238812, #72238814, #72238816 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296465);