Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352341 |
| 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 | |
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:
(1,2,8,5)(3,15,21,48)(4,22,26,11)(6,12,33,28)(7,35,38,24)(9,14,31,30)(10,40,17,19)(13,44,29,39)(16,46,50,55)(18,47,56,64)(20,45,54,60)(23,57,32,27)(25,58,37,34)(36,43,63,53)(41,42,51,52)(49,62,61,59);(1,3)(2,10)(4,16)(5,17)(6,18)(7,36)(8,21)(9,20)(11,41)(12,42)(13,45)(14,43)(15,35)(19,39)(22,51)(23,49)(24,48)(25,59)(26,50)(27,47)(28,52)(29,60)(30,53)(31,54)(32,61)(33,56)(34,55)(37,62)(38,63)(40,44)(46,58)(57,64);(1,4,9,27)(2,11,14,32)(3,16,20,47)(5,22,30,23)(6,24,34,29)(7,37,39,28)(8,26,31,57)(10,41,43,61)(12,38,25,44)(13,33,35,58)(15,46,45,56)(17,51,53,49)(18,48,55,60)(19,52,36,62)(21,50,54,64)(40,42,63,59);(1,5,8,2)(3,17,21,10)(4,23,26,32)(6,12,33,28)(7,29,38,13)(9,30,31,14)(11,27,22,57)(15,19,48,40)(16,49,50,61)(18,42,56,52)(20,53,54,43)(24,44,35,39)(25,58,37,34)(36,60,63,45)(41,47,51,64)(46,62,55,59);(1,6,31,58)(2,12,30,37)(3,18,54,46)(4,24,57,13)(5,28,14,25)(7,32,44,22)(8,33,9,34)(10,42,53,62)(11,38,23,39)(15,47,60,50)(16,48,64,45)(17,52,43,59)(19,41,63,49)(20,55,21,56)(26,35,27,29)(36,61,40,51)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=i,e^2=hi,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259632);