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 | #63061 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2537 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)(4,20)(5,11)(6,12)(7,13)(8,14)(9,18)(10,21)(16,26)(17,27)(19,32)(22,28)(23,30)(24,31)(25,29)(33,34,39,43)(35,44,47,55)(36,48,50,41)(37,42,38,52)(40,53,56,46)(45,60,63,59)(49,58,51,57)(54,62,64,61);(1,3,8,19)(2,9,14,27)(4,16,22,5)(6,17,7,18)(10,26,29,11)(12,15,13,32)(20,30,28,31)(21,24,25,23)(33,35)(34,40)(36,45)(37,46)(38,53)(39,47)(41,54)(42,55)(43,56)(44,52)(48,64)(49,61)(50,63)(51,62)(57,59)(58,60);(1,4)(2,10)(3,16)(5,19)(6,21)(7,25)(8,22)(9,26)(11,27)(12,28)(13,20)(14,29)(15,31)(17,24)(18,23)(30,32)(33,34)(35,40)(36,41)(37,42)(38,52)(39,43)(44,53)(45,54)(46,55)(47,56)(48,50)(49,57)(51,58)(59,61)(60,62)(63,64);(1,5,8,16)(2,11,14,26)(3,4,19,22)(6,24,7,23)(9,10,27,29)(12,31,13,30)(15,20,32,28)(17,25,18,21)(33,36,39,50)(34,41,43,48)(35,45,47,63)(37,51,38,49)(40,54,56,64)(42,58,52,57)(44,59,55,60)(46,62,53,61);(1,6,8,7)(2,12,14,13)(3,17,19,18)(4,21,22,25)(5,23,16,24)(9,15,27,32)(10,28,29,20)(11,30,26,31)(33,37,39,38)(34,42,43,52)(35,46,47,53)(36,49,50,51)(40,55,56,44)(41,57,48,58)(45,61,63,62)(54,59,64,60)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=h,e^2=h,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, 6259137);