Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453836 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16146 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,8,14,7,12)(3,15,18,9,20,32,19,21)(4,17,23,10,24,28,22,29)(5,16,25,11,27,30,26,31);(1,3)(2,9,14,21)(4,11)(5,10,27,29)(6,18)(7,19)(8,20)(12,15,13,32)(16,22)(17,25,28,26)(23,30)(24,31);(1,4,6,22,8,24,7,23)(2,10,12,28,14,29,13,17)(3,16,18,31,20,30,19,11)(5,21,25,32,27,9,26,15);(1,5,6,25,8,27,7,26)(2,11,13,30,14,31,12,16)(3,17,19,29,20,28,18,10)(4,21,22,32,24,9,23,15)Polycyclic presentation:
2^9|a^2=efg,b^2=fh,c^2=e,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361032);