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:
(1,2,6,11,7,12,20,40)(3,13,16,31,17,38,27,8)(4,18,22,39,23,9,28,34)(5,10,25,36,26,37,44,57)(14,41,43,29,19,48,46,32)(15,35,24,53,45,61,54,30)(21,47,51,62,52,33,55,60)(42,59,63,56,49,64,50,58);(1,3,7,17)(2,8,12,31)(4,19,23,14)(5,24,26,54)(6,27,20,16)(9,32,18,29)(10,35,37,61)(11,38,40,13)(15,44,45,25)(21,50,52,63)(22,43,28,46)(30,57,53,36)(33,59,47,64)(34,48,39,41)(42,51,49,55)(56,60,58,62);(1,4,20,22,7,23,6,28)(2,9,11,39,12,18,40,34)(3,14,16,46,17,19,27,43)(5,21,44,51,26,52,25,55)(8,29,38,48,31,32,13,41)(10,33,36,62,37,47,57,60)(15,42,24,50,45,49,54,63)(30,56,61,64,53,58,35,59);(1,5,20,44,7,26,6,25)(2,10,11,36,12,37,40,57)(3,15,16,24,17,45,27,54)(4,21,22,51,23,52,28,55)(8,30,38,61,31,53,13,35)(9,33,39,62,18,47,34,60)(14,42,46,50,19,49,43,63)(29,56,48,64,32,58,41,59);(1,3)(2,8)(4,14)(5,15)(6,27)(7,17)(9,29)(10,30)(11,38)(12,31)(13,40)(16,20)(18,32)(19,23)(21,42)(22,46)(24,44)(25,54)(26,45)(28,43)(33,56)(34,41)(35,57)(36,61)(37,53)(39,48)(47,58)(49,52)(50,51)(55,63)(59,60)(62,64)Polycyclic presentation:
2^9|a^2=f,b^2=i,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=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, 7118952);