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 | |
2657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,4)(3,13,19,42)(5,11,26,21)(6,27,29,39)(7,10,23,20)(9,31,14,16)(12,37,22,30)(15,41,47,63)(17,40,18,38)(24,52,54,61)(25,35,51,49)(28,32,48,34)(33,56,43,44)(36,59,50,55)(45,62,46,60)(53,57,64,58);(1,3)(2,9)(4,14)(5,15)(6,28)(7,18)(8,19)(10,32)(11,33)(12,38)(13,27)(16,30)(17,23)(20,34)(21,43)(22,40)(24,53)(25,46)(26,47)(29,48)(31,37)(35,57)(36,60)(39,42)(41,52)(44,55)(45,51)(49,58)(50,62)(54,64)(56,59)(61,63);(2,10)(4,20)(6,29)(9,32)(11,35)(12,39)(13,40)(14,34)(16,31)(21,49)(22,27)(24,54)(28,48)(30,37)(33,57)(36,61)(38,42)(41,62)(43,58)(44,56)(50,52)(53,64)(55,59)(60,63);(1,4,8,2)(3,14,19,9)(5,11,26,21)(6,22,29,12)(7,20,23,10)(13,16,42,31)(15,33,47,43)(17,32,18,34)(24,36,54,50)(25,35,51,49)(27,30,39,37)(28,40,48,38)(41,56,63,44)(45,58,46,57)(52,59,61,55)(53,60,64,62);(1,5,23,51)(2,11,20,49)(3,15,17,45)(4,21,10,35)(6,24,37,59)(7,25,8,26)(9,33,34,58)(12,36,39,61)(13,41,40,62)(14,43,32,57)(16,44,48,64)(18,46,19,47)(22,50,27,52)(28,53,31,56)(29,54,30,55)(38,60,42,63)Polycyclic presentation:
2^9|a^2=i,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, 6259631);