Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
771 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)(3,6)(4,8)(5,7)(9,10,14,21,15,20,34,19)(11,22,27,35,26,36,25,16)(12,28,29,38,24,17,30,39)(13,31,33,37,32,40,23,18);(1,3,5,4)(2,6,7,8)(9,11,15,26)(10,16,20,35)(12,13,24,32)(14,27,34,25)(17,37,28,18)(19,36,21,22)(23,29,33,30)(31,39,40,38);(1,4,5,3)(2,6,7,8)(9,12,15,24)(10,17,20,28)(11,23,26,33)(13,25,32,27)(14,29,34,30)(16,31,35,40)(18,22,37,36)(19,38,21,39);(9,13,14,33,15,32,34,23)(10,18,21,31,20,37,19,40)(11,24,27,30,26,12,25,29)(16,17,22,39,35,28,36,38)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284452);