Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563317 |
| 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 | |
2592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,15,20,36)(4,21,24,12)(5,25,29,56)(6,10,30,40)(7,31,32,13)(9,19,37,18)(11,41,43,28)(16,33,48,35)(17,34,26,57)(22,44,47,38)(23,53,55,39)(27,51,59,42)(45,49,62,50)(46,63,58,64)(52,60,54,61);(1,3,7,18,8,20,32,19)(2,9,13,15,14,37,31,36)(4,22,6,16,24,47,30,48)(5,26,28,50,29,17,41,49)(10,38,21,33,40,44,12,35)(11,34,25,45,43,57,56,62)(23,54,27,58,55,52,59,46)(39,63,51,60,53,64,42,61);(1,4,7,6,8,24,32,30)(2,10,13,21,14,40,31,12)(3,16,19,22,20,48,18,47)(5,23,28,27,29,55,41,59)(9,33,36,38,37,35,15,44)(11,39,25,51,43,53,56,42)(17,46,50,52,26,58,49,54)(34,60,62,63,57,61,45,64);(1,5,7,28,8,29,32,41)(2,11,13,25,14,43,31,56)(3,17,19,50,20,26,18,49)(4,23,6,27,24,55,30,59)(9,34,36,62,37,57,15,45)(10,39,21,51,40,53,12,42)(16,46,22,52,48,58,47,54)(33,60,38,63,35,61,44,64);(1,6,7,24,8,30,32,4)(2,12,13,10,14,21,31,40)(3,16,18,47,20,48,19,22)(5,27,28,55,29,59,41,23)(9,35,15,38,37,33,36,44)(11,42,25,39,43,51,56,53)(17,46,49,54,26,58,50,52)(34,61,45,63,57,60,62,64)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=g,e^2=f,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,e^c=ei,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470514);