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 | #63047 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2570 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)(3,13,16,26)(4,9,17,23)(5,11,6,10)(8,20,22,15)(14,27,28,32)(18,25,19,24)(21,30,31,29)(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)(2,8)(4,14)(5,20)(6,15)(7,16)(9,21)(10,26)(11,13)(12,22)(17,28)(18,30)(19,29)(23,31)(24,32)(25,27)(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);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,21)(10,13)(11,26)(12,22)(17,28)(18,29)(19,30)(23,31)(24,27)(25,32)(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);(1,4,7,17)(2,9,12,23)(3,14,16,28)(5,18,6,19)(8,21,22,31)(10,24,11,25)(13,27,26,32)(15,29,20,30)(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);(4,17)(9,23)(14,28)(18,19)(21,31)(24,25)(27,32)(29,30)(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=h,d^2=h,e^2=i,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, 6259202);