Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380594 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14271 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,18,41,19,9,28,31)(4,10,22,35,23,34,27,40)(5,11,26,39,24,37,25,38)(16,42,45,32,46,29,20,49)(17,43,48,62,47,30,54,57)(21,33,52,59,51,36,53,61)(44,56,64,58,63,55,50,60);(1,3,6,18,8,19,7,28)(2,9,13,31,12,15,14,41)(4,20,22,46,23,45,27,16)(5,17,25,48,24,47,26,54)(10,32,35,42,34,49,40,29)(11,30,38,57,37,43,39,62)(21,50,52,63,51,64,53,44)(33,58,59,56,36,60,61,55);(1,4,8,23)(2,10,12,34)(3,16,19,46)(5,21,24,51)(6,27,7,22)(9,29,15,42)(11,36,37,33)(13,40,14,35)(17,44,47,63)(18,45,28,20)(25,53,26,52)(30,56,43,55)(31,49,41,32)(38,59,39,61)(48,64,54,50)(57,58,62,60);(1,5)(2,11)(3,17)(4,21)(6,25)(7,26)(8,24)(9,30)(10,33)(12,37)(13,38)(14,39)(15,43)(16,44)(18,48)(19,47)(20,50)(22,52)(23,51)(27,53)(28,54)(29,55)(31,57)(32,58)(34,36)(35,59)(40,61)(41,62)(42,56)(45,64)(46,63)(49,60)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,h^b=hi,d^c=de,g^c=gi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287793);