Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473597 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16873 |
| 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 | |
705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,4,7,12,8,11)(3,13,16,14,18,19,17,9)(5,21,22,26,24,20,23,10)(15,25,29,32,30,28,31,27);(1,3,8,16,7,18,6,17)(2,9,11,13,12,14,4,19)(5,15,22,31,24,30,23,29)(10,25,21,27,26,28,20,32);(1,4,8,12,7,11,6,2)(3,14,16,13,18,9,17,19)(5,10,22,20,24,26,23,21)(15,27,31,25,30,32,29,28);(1,5,8,22,7,24,6,23)(2,10,4,20,12,26,11,21)(3,15,16,31,18,30,17,29)(9,25,19,32,14,28,13,27)Polycyclic presentation:
2^9|a^2=e,b^2=egh,c^2=ef,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380793);