Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351101 |
| 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 | #3460 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,11)(3,12,15,22)(4,9,17,25)(5,10,6,19)(8,20,23,14)(13,27,30,32)(16,24,18,26)(21,28,31,29)(33,34)(35,45)(36,42)(37,44)(38,41)(39,43)(40,47)(46,59)(48,55)(49,53)(50,58)(51,57)(52,56)(54,61)(60,63)(62,64);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,21)(10,22)(11,23)(12,19)(16,28)(17,30)(18,29)(24,27)(25,31)(26,32)(33,35,39,49)(34,40,43,55)(36,46,50,60)(37,48,38,47)(41,53,44,45)(42,54,58,64)(51,62,52,61)(56,63,57,59);(1,4,7,17)(2,9,11,25)(3,13,15,30)(5,18,6,16)(8,21,23,31)(10,26,19,24)(12,27,22,32)(14,29,20,28)(34,41)(36,50)(37,38)(40,53)(42,57)(43,44)(45,55)(46,60)(47,48)(54,59)(56,58)(63,64);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,16,17,18)(8,22,23,12)(9,24,25,26)(13,28,30,29)(21,27,31,32)(33,36,39,50)(34,42,43,58)(35,46,49,60)(37,51,38,52)(40,54,55,64)(41,56,44,57)(45,59,53,63)(47,61,48,62);(2,11)(8,23)(9,25)(10,19)(12,22)(21,31)(24,26)(27,32)(34,43)(36,50)(40,55)(41,44)(45,53)(46,60)(51,52)(61,62)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258534);