Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563313 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #59126 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,17)(4,13)(5,26)(6,11)(7,15)(8,14)(9,16)(10,20)(12,30)(18,45)(19,48)(21,43)(22,36)(23,52)(24,39)(25,38)(27,34)(28,40)(29,42)(31,41)(32,35)(33,47)(37,55)(44,49)(46,64)(50,63)(51,61)(53,59)(54,62)(56,58)(57,60);(1,3,7,20,9,21,8,32)(2,10,14,17,16,35,15,43)(4,22,24,47,25,51,6,18)(5,27,29,58,31,19,30,49)(11,36,38,45,39,61,13,33)(12,34,41,44,42,48,26,56)(23,53,54,46,55,50,28,57)(37,60,62,63,52,64,40,59);(1,4,7,24,9,25,8,6)(2,11,14,38,16,39,15,13)(3,18,32,51,21,47,20,22)(5,23,29,54,31,55,30,28)(10,33,43,61,35,45,17,36)(12,37,41,62,42,52,26,40)(19,46,58,53,27,57,49,50)(34,59,56,64,48,63,44,60);(1,5,8,30,9,31,7,29)(2,12,15,26,16,42,14,41)(3,19,20,49,21,27,32,58)(4,23,6,28,25,55,24,54)(10,34,17,44,35,48,43,56)(11,37,13,40,39,52,38,62)(18,46,22,50,47,57,51,53)(33,59,36,60,45,63,61,64);(1,6,8,25,9,24,7,4)(2,13,15,39,16,38,14,11)(3,18,32,51,21,47,20,22)(5,28,30,55,31,54,29,23)(10,33,43,61,35,45,17,36)(12,40,26,52,42,62,41,37)(19,46,58,53,27,57,49,50)(34,59,56,64,48,63,44,60)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=g,e^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470511);