Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349370 |
| 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 | #60961 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2573 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)(5,11)(6,20)(7,10)(8,15)(13,26)(14,22)(16,25)(17,24)(18,23)(19,30)(21,29)(27,32)(28,31)(33,34)(35,45)(36,50)(37,42)(38,43)(39,44)(40,49)(41,52)(46,60)(47,55)(48,56)(51,58)(53,57)(54,62)(59,61)(63,64);(1,3,7,16)(2,8,10,22)(4,13,17,27)(5,15,6,14)(9,21,24,31)(11,25,20,12)(18,29,19,28)(23,32,30,26)(33,35,39,48)(34,40,44,55)(36,46,53,62)(37,47,38,49)(41,54,58,60)(42,45,43,56)(50,64,57,59)(51,61,52,63);(1,4,7,17)(2,9,10,24)(3,13,16,27)(5,18,6,19)(8,21,22,31)(11,23,20,30)(12,26,25,32)(14,28,15,29)(33,36,39,53)(34,41,44,58)(35,46,48,62)(37,52,38,51)(40,54,55,60)(42,57,43,50)(45,59,56,64)(47,63,49,61);(2,10)(4,17)(8,22)(11,20)(12,25)(13,27)(18,19)(28,29)(33,37,39,38)(34,42,44,43)(35,47,48,49)(36,51,53,52)(40,45,55,56)(41,50,58,57)(46,61,62,63)(54,64,60,59);(1,5,7,6)(2,11,10,20)(3,14,16,15)(4,18,17,19)(8,12,22,25)(9,23,24,30)(13,28,27,29)(21,26,31,32)(35,48)(40,55)(45,56)(46,62)(47,49)(54,60)(59,64)(61,63)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256628);