Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7535051 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #59041 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1724 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)(4,23)(5,28)(6,11)(7,15)(8,14)(9,16)(10,20)(12,31)(13,27)(18,34)(19,35)(21,38)(22,37)(24,46)(25,55)(26,41)(29,47)(30,53)(32,45)(33,44)(36,50)(39,49)(40,57)(42,51)(43,58)(48,62)(52,61)(54,59)(56,63)(60,64);(1,3,7,20,9,22,8,21)(2,10,14,17,16,38,15,37)(4,24,6,18,27,49,26,50)(5,29,31,19,32,51,33,52)(11,39,23,34,41,46,13,36)(12,42,28,35,44,47,45,61)(25,56,30,54,57,48,58,64)(40,60,53,62,55,59,43,63);(1,4,8,26,9,27,7,6)(2,11,15,13,16,41,14,23)(3,18,20,49,22,50,21,24)(5,25,33,58,32,57,31,30)(10,34,17,46,38,36,37,39)(12,40,45,43,44,55,28,53)(19,48,51,64,52,56,29,54)(35,59,47,63,61,60,42,62);(1,5,7,31,9,32,8,33)(2,12,14,28,16,44,15,45)(3,19,20,51,22,52,21,29)(4,25,6,30,27,57,26,58)(10,35,17,47,38,61,37,42)(11,40,23,53,41,55,13,43)(18,48,49,64,50,56,24,54)(34,59,46,63,36,60,39,62);(1,6,7,27,9,26,8,4)(2,13,14,11,16,23,15,41)(3,18,20,49,22,50,21,24)(5,30,31,57,32,58,33,25)(10,36,17,39,38,34,37,46)(12,43,28,40,44,53,45,55)(19,48,51,64,52,56,29,54)(35,60,47,62,61,59,42,63)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=f,e^2=f,f^2=i,g^2=i,h^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^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7442248);