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 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,17,22)(4,9,18,24)(5,11,6,10)(8,15,23,16)(14,27,28,32)(19,26,20,25)(21,29,31,30)(33,34,39,43)(35,45,48,55)(36,50,53,57)(37,44,38,42)(40,47,56,49)(41,51,58,52)(46,60,62,54)(59,63,64,61);(1,3,7,17)(2,8,12,23)(4,14,18,28)(5,16,6,15)(9,21,24,31)(10,13,11,22)(19,30,20,29)(25,27,26,32)(33,35,39,48)(34,40,43,56)(36,46,53,62)(37,47,38,49)(41,54,58,60)(42,55,44,45)(50,64,57,59)(51,61,52,63);(1,4,7,18)(2,9,12,24)(3,14,17,28)(5,19,6,20)(8,21,23,31)(10,25,11,26)(13,27,22,32)(15,29,16,30)(33,36,39,53)(34,41,43,58)(35,46,48,62)(37,52,38,51)(40,54,56,60)(42,50,44,57)(45,59,55,64)(47,63,49,61);(4,18)(9,24)(14,28)(19,20)(21,31)(25,26)(27,32)(29,30)(33,37,39,38)(34,42,43,44)(35,47,48,49)(36,51,53,52)(40,55,56,45)(41,57,58,50)(46,61,62,63)(54,59,60,64);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,18,20)(8,22,23,13)(9,25,24,26)(14,29,28,30)(21,32,31,27)(34,43)(35,48)(41,58)(42,44)(46,62)(47,49)(50,57)(61,63)Polycyclic presentation:
2^9|a^2=h,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=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, 6256877);