Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352469 |
| 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×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,14)(3,17,35,63)(4,11,25,12)(5,27,26,23)(6,29,32,31)(8,15,9,16)(10,36,45,20)(13,34,44,33)(18,47,55,48)(19,53,52,64)(21,43,22,49)(24,56,58,61)(28,42,59,41)(30,39,46,40)(37,62,38,51)(50,54,60,57);(1,3,8,21)(2,10,15,39)(4,18,26,52)(5,19,25,55)(6,30,33,20)(7,35,9,22)(11,37,23,50)(12,38,27,60)(13,43,29,17)(14,45,16,40)(24,57,59,51)(28,62,58,54)(31,63,44,49)(32,46,34,36)(41,64,56,47)(42,53,61,48);(1,2)(3,10)(4,23)(5,12)(6,31)(7,14)(8,15)(9,16)(11,26)(13,34)(17,46)(18,50)(19,38)(20,49)(21,39)(22,40)(24,42)(25,27)(28,56)(29,32)(30,63)(33,44)(35,45)(36,43)(37,52)(41,58)(47,62)(48,51)(53,57)(54,64)(55,60)(59,61);(1,4)(2,11)(3,18)(5,9)(6,24)(7,25)(8,26)(10,37)(12,14)(13,41)(15,23)(16,27)(17,47)(19,22)(20,51)(21,52)(28,34)(29,56)(30,57)(31,61)(32,58)(33,59)(35,55)(36,62)(38,45)(39,50)(40,60)(42,44)(43,64)(46,54)(48,63)(49,53);(1,5,9,4)(2,12,16,23)(3,19,22,18)(6,28,34,24)(7,26,8,25)(10,38,40,50)(11,15,27,14)(13,42,31,56)(17,48,49,64)(20,54,46,51)(21,55,35,52)(29,61,44,41)(30,62,36,57)(32,59,33,58)(37,39,60,45)(43,53,63,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259762);