Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351817 |
| 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 | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2621 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,17)(5,12)(6,11)(7,10)(8,16)(9,19)(14,22)(15,25)(18,26)(20,23)(21,24)(27,30)(28,32)(29,31)(33,34)(35,44)(36,41)(37,42)(38,52)(39,43)(40,46)(45,60)(47,55)(48,54)(49,56)(50,57)(51,58)(53,61)(59,62)(63,64);(1,3)(2,8)(4,14)(5,20)(6,16)(7,15)(9,22)(10,23)(11,25)(12,13)(17,31)(18,28)(19,30)(21,29)(24,32)(26,27)(33,35)(34,40)(36,45)(37,46)(38,47)(39,48)(41,53)(42,54)(43,55)(44,52)(49,61)(50,63)(51,62)(56,59)(57,60)(58,64);(1,4)(2,9)(3,14)(5,19)(6,21)(7,18)(8,22)(10,24)(11,17)(12,26)(13,27)(15,28)(16,29)(20,30)(23,32)(25,31)(33,36)(34,41)(35,45)(37,50)(38,49)(39,51)(40,53)(42,57)(43,58)(44,59)(46,63)(47,61)(48,62)(52,56)(54,60)(55,64);(2,10)(4,18)(8,23)(11,12)(13,25)(14,28)(19,21)(29,30)(33,37,39,38)(34,42,43,52)(35,46,48,47)(36,49,51,50)(40,54,55,44)(41,56,58,57)(45,61,62,63)(53,59,64,60);(3,15)(8,23)(13,25)(14,28)(16,20)(22,32)(27,31)(29,30)(33,37,39,38)(34,42,43,52)(35,47,48,46)(36,50,51,49)(40,44,55,54)(41,57,58,56)(45,61,62,63)(53,59,64,60)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259056);