Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351885 |
| 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 | #63064 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,14,18,20)(4,17,9,16)(5,10,23,27)(6,11,7,12)(15,28,30,32)(19,29,24,31)(21,25,22,26)(33,34,39,36)(35,43,47,56)(37,48,50,41)(38,49,53,42)(40,52,44,46)(45,59,60,57)(51,55,63,62)(54,61,58,64);(1,3)(2,9)(4,13)(5,15)(6,16)(7,17)(8,18)(10,24)(11,14)(12,20)(19,27)(21,29)(22,31)(23,30)(25,32)(26,28)(33,35)(34,40)(36,44)(37,45)(38,52)(39,47)(41,54)(42,56)(43,49)(46,53)(48,58)(50,60)(51,64)(55,59)(57,62)(61,63);(1,4,8,9)(2,3,13,18)(5,19,23,24)(6,20,7,14)(10,15,27,30)(11,16,12,17)(21,28,22,32)(25,29,26,31)(33,36,39,34)(35,44,47,40)(37,48,50,41)(38,42,53,49)(43,52,56,46)(45,58,60,54)(51,55,63,62)(57,64,59,61);(1,5)(2,10)(3,15)(4,19)(6,22)(7,21)(8,23)(9,24)(11,26)(12,25)(13,27)(14,28)(16,31)(17,29)(18,30)(20,32)(33,37)(34,41)(35,45)(36,48)(38,51)(39,50)(40,54)(42,55)(43,57)(44,58)(46,61)(47,60)(49,62)(52,64)(53,63)(56,59);(1,6,8,7)(2,11,13,12)(3,16,18,17)(4,20,9,14)(5,21,23,22)(10,25,27,26)(15,29,30,31)(19,28,24,32)(37,50)(41,48)(45,60)(51,63)(54,58)(55,62)(57,59)(61,64)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259214);