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 |
| prime radical | #62828 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,16,23)(4,18,21,25)(5,11,6,10)(8,15,24,17)(9,19,26,20)(14,22,29,32)(27,28,31,30)(33,34,39,43)(35,44,48,58)(36,41,49,55)(37,42,38,51)(40,46,54,47)(45,60,61,59)(50,56,52,57)(53,63,64,62);(1,3)(2,8)(4,14)(5,15)(6,17)(7,16)(9,22)(10,23)(11,13)(12,24)(18,31)(19,28)(20,30)(21,29)(25,27)(26,32)(33,35)(34,40)(36,45)(37,47)(38,46)(39,48)(41,53)(42,58)(43,54)(44,51)(49,61)(50,63)(52,62)(55,64)(56,60)(57,59);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,18)(11,25)(12,26)(13,27)(15,30)(16,29)(17,28)(23,31)(24,32)(33,36)(34,41)(35,45)(37,50)(38,52)(39,49)(40,53)(42,56)(43,55)(44,59)(46,62)(47,63)(48,61)(51,57)(54,64)(58,60);(1,5,7,6)(2,10,12,11)(3,15,16,17)(4,19,21,20)(8,23,24,13)(9,25,26,18)(14,28,29,30)(22,27,32,31)(36,49)(41,55)(45,61)(50,52)(53,64)(56,57)(59,60)(62,63);(3,16)(8,24)(13,23)(14,29)(15,17)(22,32)(27,31)(28,30)(33,37,39,38)(34,42,43,51)(35,46,48,47)(36,50,49,52)(40,44,54,58)(41,56,55,57)(45,62,61,63)(53,59,64,60)Polycyclic presentation:
2^9|a^2=h,d^2=hi,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, 6259058);