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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2551 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,16)(9,19)(14,22)(15,23)(17,26)(20,24)(21,25)(27,28)(29,31)(30,32)(34,39)(36,46)(37,48)(40,53)(41,56)(42,49)(43,58)(44,50)(51,55)(52,57)(54,62)(63,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,22)(10,13)(11,26)(12,23)(18,31)(19,28)(20,30)(21,29)(24,27)(25,32)(33,34)(35,40)(36,41)(37,49)(38,42)(39,48)(43,54)(44,51)(45,55)(46,57)(47,56)(50,53)(52,60)(58,64)(59,62)(61,63);(1,4,7,20)(2,9,12,25)(3,14,17,30)(5,19,6,21)(8,22,23,32)(10,24,11,18)(13,27,26,31)(15,28,16,29)(33,35,38,45)(34,40,42,55)(36,43,46,58)(37,50,48,44)(39,51,49,53)(41,54,57,64)(47,61,60,59)(52,62,56,63);(1,2)(3,8)(4,9)(5,11)(6,10)(7,12)(13,16)(14,22)(15,26)(17,23)(18,19)(20,25)(21,24)(27,29)(28,31)(30,32)(33,36)(34,41)(35,43)(37,47)(38,46)(39,52)(40,54)(42,57)(44,59)(45,58)(48,60)(49,56)(50,61)(51,62)(53,63)(55,64);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,13,23,26)(9,24,25,18)(14,28,30,29)(22,27,32,31)(36,46)(41,57)(43,58)(47,60)(52,56)(54,64)(59,61)(62,63)Polycyclic presentation:
2^9|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, 6259213);