Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7211759 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #45599 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #30498 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,28,7,13,6,12)(3,14,26,9,18,40,17,32)(4,10,27,41,22,36,21,35)(5,11,29,55,25,39,24,38)(15,43,47,30,19,42,45,33)(16,37,53,31,46,61,23,52)(20,34,54,62,51,60,50,59)(44,63,49,56,48,58,64,57);(1,3)(2,9)(4,19)(5,23)(6,26)(7,18)(8,17)(10,33)(11,37)(12,40)(13,32)(14,28)(15,22)(16,24)(20,49)(21,45)(25,53)(27,47)(29,46)(30,36)(31,38)(34,58)(35,43)(39,61)(41,42)(44,54)(48,50)(51,64)(52,55)(56,62)(57,59)(60,63);(1,4,6,27,7,22,8,21)(2,10,28,35,13,36,12,41)(3,15,26,45,18,19,17,47)(5,20,24,54,25,51,29,50)(9,30,14,42,32,33,40,43)(11,34,55,59,39,60,38,62)(16,44,53,64,46,48,23,49)(31,56,37,58,52,57,61,63);(1,5,6,24,7,25,8,29)(2,11,28,55,13,39,12,38)(3,16,26,53,18,46,17,23)(4,20,27,54,22,51,21,50)(9,31,14,37,32,52,40,61)(10,34,35,59,36,60,41,62)(15,44,45,64,19,48,47,49)(30,56,42,58,33,57,43,63);(1,3)(2,9)(4,15)(5,16)(6,26)(7,18)(8,17)(10,30)(11,31)(12,40)(13,32)(14,28)(19,22)(20,44)(21,47)(23,29)(24,53)(25,46)(27,45)(33,36)(34,56)(35,42)(37,55)(38,61)(39,52)(41,43)(48,51)(49,50)(54,64)(57,60)(58,59)(62,63)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^d=hi,f^e=fi,g^e=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7118956);