Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| 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 | C22×C4 | C25 |
| prime radical | #537 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2543 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)(3,12,15,23)(4,16,18,9)(5,10,6,20)(8,21,24,14)(13,28,31,27)(17,26,19,25)(22,30,32,29)(39,50)(51,61)(52,62)(63,64);(1,3)(2,8)(4,13)(5,14)(6,21)(7,15)(9,22)(10,23)(11,24)(12,20)(16,32)(17,29)(18,31)(19,30)(25,27)(26,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,51)(44,52)(45,50)(46,53)(47,54)(48,55)(49,56)(57,63)(58,61)(59,64)(60,62);(33,35,38,45)(34,39,42,50)(36,43,47,58)(37,44,49,60)(40,51,54,61)(41,52,56,62)(46,57,48,59)(53,63,55,64);(1,4,7,18)(2,9,11,16)(3,13,15,31)(5,19,6,17)(8,22,24,32)(10,26,20,25)(12,27,23,28)(14,30,21,29)(33,36,38,47)(34,40,42,54)(35,43,45,58)(37,48,49,46)(39,51,50,61)(41,55,56,53)(44,59,60,57)(52,64,62,63);(1,5,7,6)(2,10,11,20)(3,14,15,21)(4,17,18,19)(8,23,24,12)(9,25,16,26)(13,29,31,30)(22,27,32,28)(33,37,38,49)(34,41,42,56)(35,44,45,60)(36,46,47,48)(39,52,50,62)(40,53,54,55)(43,57,58,59)(51,63,61,64)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=hi,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260356);