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 | |
2590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8)(3,12)(5,6)(10,21)(11,18)(13,26)(14,19)(16,17)(20,23)(25,28)(27,29)(30,32)(33,34)(35,45)(36,41)(37,43)(38,52)(39,44)(40,47)(42,50)(46,61)(48,56)(49,58)(51,59)(53,57)(54,63)(55,62)(60,64);(1,2,7,11)(3,9,14,24)(4,10,15,25)(5,18,6,8)(12,23,19,20)(13,22,27,31)(16,28,17,21)(26,32,29,30)(33,35,39,48)(34,40,44,57)(36,46,50,64)(37,47,38,53)(41,54,42,55)(43,56,52,45)(49,62,51,63)(58,61,59,60);(1,3)(2,9)(4,13)(5,12)(6,19)(7,14)(8,20)(10,22)(11,24)(15,27)(16,26)(17,29)(18,23)(21,30)(25,31)(28,32)(33,36)(34,41)(35,46)(37,49)(38,51)(39,50)(40,54)(42,44)(43,58)(45,60)(47,62)(48,64)(52,59)(53,63)(55,57)(56,61);(1,4)(2,10)(3,13)(5,16)(6,17)(7,15)(8,21)(9,22)(11,25)(12,26)(14,27)(18,28)(19,29)(20,30)(23,32)(24,31)(33,36)(34,42)(35,46)(37,51)(38,49)(39,50)(40,55)(41,44)(43,58)(45,60)(47,63)(48,64)(52,59)(53,62)(54,57)(56,61);(4,15)(10,25)(13,27)(16,17)(21,28)(22,31)(26,29)(30,32)(33,37,39,38)(34,43,44,52)(35,47,48,53)(36,49,50,51)(40,56,57,45)(41,58,42,59)(46,62,64,63)(54,61,55,60)Polycyclic presentation:
2^9|b^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, 6259195);