Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380266 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14262 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,27)(3,14,17,32,18,9,29,33)(4,19,23,36,21,50,22,10)(5,11,25,39,26,37,28,38)(15,41,44,34,43,56,53,30)(16,40,47,58,48,31,46,57)(20,49,51,60,24,54,52,35)(42,61,63,59,45,64,62,55);(1,3)(2,9)(4,15)(5,16)(6,17)(7,29)(8,18)(10,34)(11,31)(12,14)(13,32)(19,41)(20,42)(21,43)(22,44)(23,53)(24,45)(25,46)(26,48)(27,33)(28,47)(30,36)(35,59)(37,40)(38,57)(39,58)(49,61)(50,56)(51,62)(52,63)(54,64)(55,60);(1,4,6,22,8,21,7,23)(2,10,13,19,12,36,27,50)(3,15,17,44,18,43,29,53)(5,24,25,52,26,20,28,51)(9,30,32,56,14,34,33,41)(11,35,38,54,37,60,39,49)(16,45,46,63,48,42,47,62)(31,55,57,61,40,59,58,64);(1,5,8,26)(2,11,12,37)(3,16,18,48)(4,20,21,24)(6,28,7,25)(9,31,14,40)(10,35,36,60)(13,39,27,38)(15,42,43,45)(17,47,29,46)(19,49,50,54)(22,52,23,51)(30,55,34,59)(32,58,33,57)(41,61,56,64)(44,63,53,62)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287465);