Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519061 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #58997 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2460 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)(3,17)(4,22)(5,28)(6,31)(7,15)(8,14)(9,16)(10,20)(11,26)(12,29)(13,32)(18,37)(19,38)(21,47)(23,49)(24,42)(25,59)(27,48)(30,39)(33,46)(34,40)(35,44)(36,45)(41,51)(43,55)(50,64)(52,62)(53,56)(54,60)(57,61)(58,63);(1,3,7,20,9,21,8,34)(2,10,14,17,16,40,15,47)(4,23,26,18,27,51,35,52)(5,19,29,6,30,32,36,33)(11,41,22,37,44,49,48,62)(12,13,28,31,45,38,39,46)(24,25,57,60,58,55,56,50)(42,54,53,59,63,64,61,43);(1,4)(2,11)(3,18)(5,24)(6,25)(7,26)(8,35)(9,27)(10,37)(12,42)(13,43)(14,22)(15,48)(16,44)(17,49)(19,50)(20,51)(21,52)(23,34)(28,53)(29,57)(30,58)(31,54)(32,60)(33,55)(36,56)(38,59)(39,61)(40,62)(41,47)(45,63)(46,64);(1,5)(2,12)(3,19)(4,24)(6,20)(7,29)(8,36)(9,30)(10,38)(11,42)(13,40)(14,28)(15,39)(16,45)(17,46)(18,50)(21,32)(22,53)(23,55)(25,51)(26,57)(27,58)(31,47)(33,34)(35,56)(37,59)(41,54)(43,62)(44,63)(48,61)(49,64)(52,60);(1,6,7,32,9,33,8,19)(2,13,14,31,16,38,15,46)(3,5,20,29,21,30,34,36)(4,25,26,60,27,55,35,50)(10,39,17,12,40,28,47,45)(11,43,22,54,44,59,48,64)(18,24,51,57,52,58,23,56)(37,61,49,42,62,53,41,63)Polycyclic presentation:
2^9|b^2=f,e^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426257);