Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2633 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,10,30)(3,16,37,61)(4,23,32,44)(5,9,15,8)(6,13,38,46)(7,34,39,29)(11,40,54,20)(12,27,59,26)(14,36,60,24)(17,47,62,55)(18,35,43,58)(19,48,51,50)(21,45,22,49)(25,28,33,31)(41,57,64,56)(42,52,63,53);(1,3)(2,11)(4,17)(5,18)(6,19)(7,35)(8,22)(9,21)(10,37)(12,41)(13,42)(14,45)(15,43)(16,34)(20,24)(23,50)(25,57)(26,53)(27,52)(28,47)(29,61)(30,54)(31,55)(32,62)(33,56)(36,40)(38,51)(39,58)(44,48)(46,63)(49,60)(59,64);(1,4,9,27)(2,12,15,44)(3,17,21,52)(5,23,30,59)(6,24,33,7)(8,26,10,32)(11,41,43,48)(13,29,31,60)(14,46,34,28)(16,47,45,63)(18,50,54,64)(19,20,56,35)(22,53,37,62)(25,39,38,36)(40,57,58,51)(42,61,55,49);(1,5,8,2)(3,18,22,11)(4,23,26,12)(6,31,25,46)(7,29,36,14)(9,30,10,15)(13,38,28,33)(16,20,49,58)(17,50,53,41)(19,55,57,63)(21,54,37,43)(24,60,39,34)(27,59,32,44)(35,61,40,45)(42,51,47,56)(48,52,64,62);(1,6,8,25)(2,13,5,28)(3,19,22,57)(4,24,26,39)(7,32,36,27)(9,33,10,38)(11,42,18,47)(12,29,23,14)(15,31,30,46)(16,48,49,64)(17,20,53,58)(21,56,37,51)(34,44,60,59)(35,62,40,52)(41,61,50,45)(43,55,54,63)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260464);