Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6369070 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,15)(4,10)(5,11)(6,13)(7,12)(8,14)(9,17)(16,30)(18,26)(19,25)(20,32)(21,29)(22,28)(23,27)(24,31)(33,34,39,45)(35,40,48,56)(36,42,50,58)(37,44,38,43)(41,52,57,49)(46,54,62,63)(47,59,51,55)(53,60,64,61);(1,3)(2,9)(4,8)(5,20)(6,17)(7,19)(10,13)(11,26)(12,15)(14,25)(16,21)(18,23)(22,31)(24,27)(28,30)(29,32)(33,35,39,48)(34,40,45,56)(36,46,50,62)(37,51,38,47)(41,53,57,64)(42,54,58,63)(43,59,44,55)(49,61,52,60);(1,4,8,3)(2,10,14,15)(5,18,22,16)(6,17,7,19)(9,12,25,13)(11,26,28,30)(20,21,31,23)(24,27,32,29)(33,36)(34,41)(35,46)(37,49)(38,52)(39,50)(40,53)(42,43)(44,58)(45,57)(47,60)(48,62)(51,61)(54,59)(55,63)(56,64);(1,5)(2,11)(3,16)(4,18)(6,21)(7,23)(8,22)(9,24)(10,26)(12,27)(13,29)(14,28)(15,30)(17,31)(19,20)(25,32)(33,36)(34,42)(35,46)(37,52)(38,49)(39,50)(40,54)(41,43)(44,57)(45,58)(47,61)(48,62)(51,60)(53,59)(55,64)(56,63);(1,6,8,7)(2,12,14,13)(3,17,4,19)(5,21,22,23)(9,15,25,10)(11,27,28,29)(16,31,18,20)(24,30,32,26)(33,37,39,38)(34,43,45,44)(35,47,48,51)(36,49,50,52)(40,55,56,59)(41,42,57,58)(46,60,62,61)(53,63,64,54)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276343);