Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574107 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,16)(3,17,6,13)(4,22,27,41)(5,12,32,47)(7,15,8,14)(10,21,37,20)(11,34,42,26)(18,50,25,35)(19,49,33,48)(23,40,52,38)(24,43,57,63)(28,59,56,39)(29,44,54,36)(30,46,31,45)(51,64,58,60)(53,62,55,61);(1,3,8,21,9,6,7,20)(2,10,15,13,16,37,14,17)(4,23,34,25,27,52,26,18)(5,19,31,29,32,33,30,54)(11,38,41,35,42,40,22,50)(12,36,46,48,47,44,45,49)(24,55,28,58,57,53,56,51)(39,62,63,60,59,61,43,64);(1,4)(2,11)(3,18)(5,28)(6,25)(7,26)(8,34)(9,27)(10,35)(12,43)(13,40)(14,22)(15,41)(16,42)(17,38)(19,53)(20,52)(21,23)(24,30)(29,51)(31,57)(32,56)(33,55)(36,61)(37,50)(39,46)(44,62)(45,59)(47,63)(48,64)(49,60)(54,58);(1,5)(2,12)(3,19)(4,24)(6,33)(7,30)(8,31)(9,32)(10,36)(11,39)(13,48)(14,45)(15,46)(16,47)(17,49)(18,51)(20,54)(21,29)(22,43)(23,55)(25,58)(26,56)(27,57)(28,34)(35,60)(37,44)(38,62)(40,61)(41,63)(42,59)(50,64)(52,53);(1,6)(2,13)(3,9)(4,25)(5,29)(7,21)(8,20)(10,14)(11,40)(12,44)(15,37)(16,17)(18,27)(19,30)(22,35)(23,26)(24,53)(28,51)(31,33)(32,54)(34,52)(36,47)(38,42)(39,60)(41,50)(43,62)(45,48)(46,49)(55,57)(56,58)(59,64)(61,63)Polycyclic presentation:
2^9|a^2=i,b^2=fi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481305);