Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #442256 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15706 |
| 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:
(1,2,7,12,8,13,6,23)(3,14,4,9,17,10,16,25)(5,19,21,11,22,28,20,29)(15,24,18,26,31,32,30,27);(1,3,7,4,8,17,6,16)(2,9,12,10,13,25,23,14)(5,18,21,31,22,30,20,15)(11,27,28,24,29,26,19,32);(1,4,6,17,8,16,7,3)(2,10,12,9,13,14,23,25)(5,18,20,31,22,30,21,15)(11,26,28,24,29,27,19,32);(1,5,7,21,8,22,6,20)(2,11,12,28,13,29,23,19)(3,15,16,30,17,31,4,18)(9,24,14,27,25,32,10,26)Polycyclic presentation:
2^9|a^2=eg,b^2=eg,c^2=e,d^2=f,e^2=i,f^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,e^b=ei,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, 349452);