Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,4,3)(5,6,11,16,12,18,10,17)(7,19,23,37,24,44,22,13)(8,25,28,41,29,40,34,14)(9,30,32,43,33,42,31,15)(20,35,47,60,48,61,26,38)(21,49,51,64,52,36,50,45)(27,39,55,53,56,63,58,57)(46,62,65,59,66,68,54,67);(1,3,4,2)(5,7,11,23,12,24,10,22)(6,13,17,44,18,37,16,19)(8,26,34,48,29,47,28,20)(9,21,32,51,33,52,31,50)(14,38,25,35,41,60,40,61)(15,36,42,64,43,49,30,45)(27,54,58,66,56,65,55,46)(39,62,53,59,63,68,57,67);(5,8,10,28,12,29,11,34)(6,14,17,25,18,41,16,40)(7,20,23,26,24,48,22,47)(9,27,31,55,33,56,32,58)(13,35,19,38,37,61,44,60)(15,39,42,53,43,63,30,57)(21,46,51,54,52,66,50,65)(36,59,45,62,49,67,64,68);(5,9,11,32,12,33,10,31)(6,15,17,42,18,43,16,30)(7,21,23,51,24,52,22,50)(8,27,34,58,29,56,28,55)(13,36,44,64,37,49,19,45)(14,39,25,53,41,63,40,57)(20,46,26,54,48,66,47,65)(35,59,60,68,61,67,38,62)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=g,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275865);