Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #443359 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15757 |
| 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 | |
713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,13,6,12)(3,14,16,9,4,18,17,26)(5,20,24,29,23,28,22,10)(15,30,21,27,19,25,31,32);(1,3,7,16,8,4,6,17)(2,9,11,18,13,26,12,14)(5,21,24,19,23,31,22,15)(10,27,20,25,29,32,28,30);(1,4,8,3)(2,9)(5,19,23,15)(6,17,7,16)(10,25)(11,14)(12,18)(13,26)(20,27)(21,22,31,24)(28,32)(29,30);(1,5,6,22,8,23,7,24)(2,10,12,28,13,29,11,20)(3,15,16,21,4,19,17,31)(9,25,18,32,26,30,14,27)Polycyclic presentation:
2^9|a^2=efg,b^2=efg,c^2=ef,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,e^b=ei,f^b=fi,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, 350555);