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 | |
2634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,30)(3,16,19,46)(4,13,29,38)(5,12,31,24)(6,11,23,43)(7,14,8,15)(10,32,34,17)(18,45,50,63)(20,42,21,47)(22,39,57,55)(25,37,54,53)(26,40,27,41)(28,35,44,36)(33,58,49,48)(51,61,52,64)(56,59,62,60);(1,3)(2,10)(4,17)(5,18)(6,28)(7,20)(8,21)(9,19)(11,16)(12,33)(13,42)(14,35)(15,36)(22,48)(23,44)(24,49)(25,56)(26,51)(27,52)(29,32)(30,34)(31,50)(37,45)(38,47)(39,61)(40,59)(41,60)(43,46)(53,63)(54,62)(55,64)(57,58);(1,4,9,29)(2,11,30,43)(3,17,19,32)(5,22,31,57)(6,8,23,7)(10,16,34,46)(12,37,24,53)(13,15,38,14)(18,48,50,58)(20,28,21,44)(25,27,54,26)(33,45,49,63)(35,42,36,47)(39,41,55,40)(51,56,52,62)(59,61,60,64);(1,2,8,15)(3,10,21,36)(4,11,23,38)(5,24,27,40)(6,13,29,43)(7,14,9,30)(12,26,41,31)(16,44,47,17)(18,49,52,59)(19,34,20,35)(22,53,54,39)(25,55,57,37)(28,42,32,46)(33,51,60,50)(45,56,64,58)(48,63,62,61);(1,5,8,27)(2,12,15,41)(3,18,21,52)(4,22,23,54)(6,25,29,57)(7,26,9,31)(10,33,36,60)(11,37,38,55)(13,39,43,53)(14,40,30,24)(16,45,47,64)(17,48,44,62)(19,50,20,51)(28,56,32,58)(34,49,35,59)(42,61,46,63)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,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, 6260494);