Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| 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 | |
884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,19,15,20,13,18)(11,21,25,32,23,36,24,16)(12,26,29,35,30,34,28,17)(22,38,40,33,39,31,27,37);(1,3,4,8)(2,5,6,7)(9,11)(10,16,20,32)(12,27,30,40)(13,24)(14,25)(15,23)(17,33)(18,36,19,21)(22,28,39,29)(26,31)(34,38)(35,37);(9,12,13,28,15,30,14,29)(10,17,19,26,20,35,18,34)(11,22,24,27,23,39,25,40)(16,31,21,37,32,38,36,33);(2,6)(5,7)(10,18,20,19)(11,23)(16,21,32,36)(17,34,35,26)(22,39)(24,25)(27,40)(31,37,38,33)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=f,d^2=fh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72268618 |
| maximal normal subgroup of | #72268618 |
| maximal quotient of | #72268631 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286584);