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 | |
1761 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)(3,13,17,8)(4,18)(5,10)(6,25)(7,11)(9,21)(12,27)(14,39,41,55)(15,37,45,29)(16,30,47,38)(19,48)(20,49)(22,53)(23,36)(24,34)(26,35)(28,42,31,58)(32,52)(33,51)(40,61,43,64)(44,56,46,57)(50,60)(54,63,59,62);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,31)(10,29)(11,13)(12,30)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,38)(26,46)(27,47)(28,33)(32,59)(34,37)(35,56)(36,57)(48,61)(49,55)(50,62)(51,58)(52,63)(53,64)(54,60);(1,4,6,21,7,20,27,51)(2,9,12,18,11,33,25,49)(3,14,16,42,17,41,47,58)(5,22,23,52,24,19,26,50)(8,28,30,55,13,31,38,39)(10,32,35,53,34,60,36,48)(15,43,44,63,45,40,46,62)(29,54,56,61,37,59,57,64);(1,5,7,24)(2,10,11,34)(3,15,17,45)(4,19,20,22)(6,26,27,23)(8,29,13,37)(9,32,33,60)(12,36,25,35)(14,40,41,43)(16,46,47,44)(18,48,49,53)(21,52,51,50)(28,54,31,59)(30,57,38,56)(39,61,55,64)(42,63,58,62)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=i,f^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, 287463);