Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349612 |
| 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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2603 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)(6,19)(7,11)(8,14)(13,27)(15,22)(16,24)(17,25)(18,26)(20,23)(21,30)(28,31)(29,32)(33,34)(35,45)(36,50)(37,44)(38,43)(39,42)(40,47)(41,52)(46,60)(48,55)(49,56)(51,57)(53,58)(54,61)(59,62)(63,64);(1,3,7,15)(2,8,11,23)(4,13,17,29)(5,14,6,20)(9,21,25,31)(10,22,19,12)(16,28,18,30)(24,27,26,32)(33,35,39,49)(34,40,42,55)(36,46,51,61)(37,48,38,47)(41,54,58,60)(43,56,44,45)(50,64,57,59)(52,63,53,62);(1,4)(2,9)(3,13)(5,18)(6,16)(7,17)(8,21)(10,26)(11,25)(12,27)(14,30)(15,29)(19,24)(20,28)(22,32)(23,31)(33,36)(34,41)(35,46)(37,52)(38,53)(39,51)(40,54)(42,58)(43,50)(44,57)(45,59)(47,62)(48,63)(49,61)(55,60)(56,64);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,16,17,18)(8,22,23,12)(9,24,25,26)(13,28,29,30)(21,27,31,32)(34,42)(36,51)(40,55)(43,44)(45,56)(46,61)(52,53)(62,63);(2,11)(3,15)(9,25)(10,19)(13,29)(14,20)(24,26)(28,30)(33,37,39,38)(34,43,42,44)(35,47,49,48)(36,52,51,53)(40,45,55,56)(41,50,58,57)(46,62,61,63)(54,59,60,64)Polycyclic presentation:
2^9|b^2=h,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 6256869);