Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7211754 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,11,7,12,19,36)(3,13,16,30,17,39,26,8)(4,9,21,33,22,34,27,40)(5,10,24,37,25,38,45,56)(14,42,44,31,18,41,47,28)(15,35,23,52,46,61,53,29)(20,32,50,59,51,60,54,62)(43,63,64,57,48,58,49,55);(1,3,7,17)(2,8,12,30)(4,18,22,14)(5,23,25,53)(6,26,19,16)(9,31,34,28)(10,35,38,61)(11,39,36,13)(15,45,46,24)(20,49,51,64)(21,44,27,47)(29,56,52,37)(32,58,60,63)(33,42,40,41)(43,50,48,54)(55,59,57,62);(1,4,19,21,7,22,6,27)(2,9,11,40,12,34,36,33)(3,14,16,47,17,18,26,44)(5,20,45,50,25,51,24,54)(8,28,39,42,30,31,13,41)(10,32,37,62,38,60,56,59)(15,43,23,49,46,48,53,64)(29,55,61,63,52,57,35,58);(1,5,6,24,7,25,19,45)(2,10,36,56,12,38,11,37)(3,15,26,53,17,46,16,23)(4,20,27,54,22,51,21,50)(8,29,13,35,30,52,39,61)(9,32,33,59,34,60,40,62)(14,43,44,64,18,48,47,49)(28,55,41,58,31,57,42,63);(1,3)(2,8)(4,14)(5,15)(6,26)(7,17)(9,28)(10,29)(11,39)(12,30)(13,36)(16,19)(18,22)(20,43)(21,47)(23,45)(24,53)(25,46)(27,44)(31,34)(32,55)(33,41)(35,56)(37,61)(38,52)(40,42)(48,51)(49,50)(54,64)(57,60)(58,59)(62,63)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,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=eh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^d=hi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7118954);