Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #443899 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15768 |
| 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 | |
744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,8,13,7,12)(3,14,16,4,17,20,18,9)(5,21,23,10,24,28,25,29)(15,19,22,32,30,26,31,27);(1,3,6,18,8,17,7,16)(2,9,11,20,13,4,12,14)(5,22,23,15,24,31,25,30)(10,27,28,26,29,32,21,19);(1,4,8,9)(2,3)(5,19,24,26)(6,20,7,14)(10,15)(11,16)(12,18)(13,17)(21,31)(22,28)(23,32,25,27)(29,30);(1,5,6,23,8,24,7,25)(2,10,11,28,13,29,12,21)(3,15,16,22,17,30,18,31)(4,19,20,32,9,26,14,27)Polycyclic presentation:
2^9|a^2=e,b^2=ef,c^2=eg,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351095);