Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349298 |
| 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 | #62838 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2591 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,15)(13,26)(14,22)(16,25)(17,23)(18,24)(19,30)(21,29)(27,32)(28,31)(33,34,39,44)(35,40,47,55)(36,48,50,58)(37,43,38,42)(41,52,56,49)(45,59,61,63)(46,57,51,54)(53,60,64,62);(1,3,7,16)(2,8,11,22)(4,13,17,27)(5,15,6,14)(9,21,23,31)(10,25,20,12)(18,29,19,28)(24,32,30,26)(33,35)(34,40)(36,45)(37,46)(38,51)(39,47)(41,53)(42,54)(43,57)(44,55)(48,63)(49,60)(50,61)(52,62)(56,64)(58,59);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,26)(14,28)(15,29)(16,27)(20,30)(22,31)(25,32)(33,36)(34,41)(35,45)(37,49)(38,52)(39,50)(40,53)(42,48)(43,58)(44,56)(46,60)(47,61)(51,62)(54,63)(55,64)(57,59);(4,17)(9,23)(13,27)(18,19)(21,31)(24,30)(26,32)(28,29)(33,37,39,38)(34,42,44,43)(35,46,47,51)(36,49,50,52)(40,54,55,57)(41,48,56,58)(45,60,61,62)(53,63,64,59);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,17,19)(8,12,22,25)(9,24,23,30)(13,28,27,29)(21,26,31,32)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256531);