Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #433831 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15493 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,25,36)(3,8,16,29,15,28,43,56)(4,9,20,33,19,32,52,61)(5,21,22,35,23,10,24,34)(13,37,17,45,39,26,47,30)(14,40,42,57,41,27,44,55)(18,48,51,62,50,31,53,60)(38,54,49,59,64,63,46,58);(1,3)(2,8)(4,17,19,47)(5,14)(6,16)(7,15)(9,30,32,45)(10,27)(11,29)(12,28)(13,20,39,52)(18,49,50,46)(21,40)(22,42)(23,41)(24,44)(25,43)(26,33,37,61)(31,59,48,58)(34,55)(35,57)(36,56)(38,51,64,53)(54,60,63,62);(1,4)(2,9)(3,13,15,39)(5,18)(6,20)(7,19)(8,26,28,37)(10,31)(11,33)(12,32)(14,38,41,64)(16,17,43,47)(21,48)(22,51)(23,50)(24,53)(25,52)(27,54,40,63)(29,30,56,45)(34,60)(35,62)(36,61)(42,49,44,46)(55,59,57,58);(1,5,7,23)(2,10,12,21)(3,14,15,41)(4,18,19,50)(6,24,25,22)(8,27,28,40)(9,31,32,48)(11,35,36,34)(13,38,39,64)(16,44,43,42)(17,46,47,49)(20,53,52,51)(26,54,37,63)(29,57,56,55)(30,58,45,59)(33,62,61,60)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341028);