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 | |
2591 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)(5,11)(6,12)(7,10)(8,15)(9,19)(14,22)(16,24)(18,25)(20,23)(21,26)(27,30)(28,32)(29,31)(33,34)(35,44)(36,41)(37,42)(38,51)(39,43)(40,46)(45,60)(47,54)(48,56)(49,58)(50,57)(52,55)(53,61)(59,62)(63,64);(1,3)(2,8)(4,14)(5,20)(6,15)(7,16)(9,22)(10,23)(11,13)(12,24)(17,31)(18,28)(19,30)(21,29)(25,27)(26,32)(33,35)(34,40)(36,45)(37,46)(38,52)(39,47)(41,53)(42,54)(43,55)(44,51)(48,61)(49,63)(50,62)(56,59)(57,64)(58,60);(1,4,7,18)(2,9,10,26)(3,14,16,28)(5,19,6,21)(8,22,23,32)(11,25,12,17)(13,27,24,31)(15,29,20,30)(33,36,39,50)(34,41,43,57)(35,45,47,62)(37,49,38,48)(40,53,55,64)(42,58,51,56)(44,59,54,60)(46,63,52,61);(2,10)(4,18)(8,23)(11,12)(13,24)(14,28)(19,21)(29,30)(33,37,39,38)(34,42,43,51)(35,46,47,52)(36,48,50,49)(40,54,55,44)(41,56,57,58)(45,61,62,63)(53,59,64,60);(1,5,7,6)(2,11,10,12)(3,15,16,20)(4,19,18,21)(8,24,23,13)(9,25,26,17)(14,29,28,30)(22,31,32,27)(35,47)(40,55)(44,54)(45,62)(46,52)(53,64)(59,60)(61,63)Polycyclic presentation:
2^9|c^2=h,d^2=i,e^2=hi,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, 6259060);