Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #386085 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14419 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60645 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1568 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)(3,13,16,8)(4,18)(5,10,23,35)(6,25)(7,11)(9,21)(12,27)(14,38,40,28)(15,29)(17,30,26,37)(19,31,47,43)(20,32,49,44)(22,33)(24,52,51,36)(34,42)(39,59)(41,60)(45,50)(46,56)(48,62,64,58)(53,55)(54,61)(57,63);(1,3)(2,8)(4,19)(5,15)(6,26)(7,16)(9,31)(10,34)(11,13)(12,37)(14,22)(17,27)(18,43)(20,46)(21,47)(23,42)(24,53)(25,30)(28,33)(29,35)(32,57)(36,55)(38,45)(39,48)(40,50)(41,51)(44,56)(49,63)(52,60)(54,62)(58,59)(61,64);(1,4,6,22,7,21,27,50)(2,9,12,33,11,18,25,45)(3,14,17,19,16,40,26,47)(5,20,24,48,23,49,51,64)(8,28,30,31,13,38,37,43)(10,32,36,58,35,44,52,62)(15,39,41,46,42,61,53,63)(29,54,55,56,34,59,60,57);(1,5,7,23)(2,10)(3,15,16,42)(4,20,21,49)(6,24,27,51)(8,29)(9,32)(11,35)(12,36)(13,34)(14,39,40,61)(17,41,26,53)(18,44)(19,46,47,63)(22,48,50,64)(25,52)(28,54)(30,55)(31,56)(33,58)(37,60)(38,59)(43,57)(45,62)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293282);