Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352341 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,6)(3,17,41,58)(4,24,27,12)(5,13,32,35)(7,25,38,49)(9,15,10,16)(11,42,20,21)(14,40,36,39)(18,52,61,57)(19,50,63,55)(22,48,23,51)(26,33,30,34)(28,31,29,47)(37,45,59,46)(43,54,53,44)(56,64,60,62);(1,3)(2,11)(4,18)(5,19)(6,20)(7,37)(8,41)(9,22)(10,23)(12,43)(13,44)(14,48)(15,45)(16,46)(17,25)(21,39)(24,53)(26,60)(27,61)(28,55)(29,50)(30,56)(31,62)(32,63)(33,52)(34,57)(35,54)(36,51)(38,59)(40,42)(47,64)(49,58);(1,4,9,28)(2,12,15,47)(3,18,22,55)(5,14,33,25)(6,24,16,31)(7,13,39,30)(8,27,10,29)(11,43,45,64)(17,19,48,52)(20,53,46,62)(21,56,37,44)(23,50,41,61)(26,38,35,40)(32,36,34,49)(42,60,59,54)(51,57,58,63);(1,5,10,34)(2,13,16,26)(3,19,23,57)(4,25,29,36)(6,35,15,30)(7,31,40,12)(8,32,9,33)(11,44,46,60)(14,27,49,28)(17,50,51,18)(20,54,45,56)(21,53,59,64)(22,52,41,63)(24,38,47,39)(37,62,42,43)(48,61,58,55);(1,6,8,2)(3,20,41,11)(4,24,27,12)(5,30,32,26)(7,36,38,14)(9,16,10,15)(13,34,35,33)(17,21,58,42)(18,53,61,43)(19,56,63,60)(22,46,23,45)(25,39,49,40)(28,31,29,47)(37,51,59,48)(44,57,54,52)(50,64,55,62)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=i,e^2=hi,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259664);