Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352701 |
| 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×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,10,25)(3,16,38,59)(4,23,39,34)(5,28,7,35)(6,13,40,27)(8,14,9,15)(11,41,50,20)(12,31,58,30)(17,46,63,61)(18,45,36,44)(19,47,64,56)(21,42,22,48)(24,32,37,60)(26,29,57,33)(43,52,51,53)(49,55,62,54);(1,3)(2,11)(4,17)(5,18)(6,19)(7,36)(8,22)(9,21)(10,38)(12,42)(13,43)(14,45)(15,44)(16,35)(20,30)(23,55)(24,49)(25,50)(26,47)(27,51)(28,59)(29,46)(31,41)(32,52)(33,61)(34,54)(37,62)(39,63)(40,64)(48,58)(53,60)(56,57);(1,4,10,39)(2,6,25,40)(3,17,38,63)(5,24,7,37)(8,27,9,13)(11,19,50,64)(12,32,58,60)(14,23,15,34)(16,46,59,61)(18,49,36,62)(20,56,41,47)(21,43,22,51)(26,30,57,31)(28,33,35,29)(42,52,48,53)(44,54,45,55);(1,5,9,31)(2,12,15,35)(3,18,21,41)(4,24,13,26)(6,32,34,29)(7,8,30,10)(11,42,44,16)(14,28,25,58)(17,49,43,47)(19,52,54,46)(20,38,36,22)(23,33,40,60)(27,57,39,37)(45,59,50,48)(51,56,63,62)(53,55,61,64);(1,6,8,23)(2,13,14,39)(3,19,22,55)(4,25,27,15)(5,29,30,60)(7,33,31,32)(9,34,10,40)(11,43,45,63)(12,24,28,57)(16,47,48,62)(17,50,51,44)(18,46,20,53)(21,54,38,64)(26,58,37,35)(36,61,41,52)(42,49,59,56)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,d^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=eh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260134);