Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357427 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60972 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1763 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)(3,12,15,22)(4,9,16,24)(5,10,6,20)(8,19,23,14)(13,26,27,32)(17,25,18,30)(21,29,31,28)(33,34,39,44)(35,45,47,56)(36,50,51,58)(37,43,38,42)(40,49,55,48)(41,52,57,53)(46,54,60,64)(59,61,63,62);(1,3,5,19,7,15,6,14)(2,8,20,22,11,23,10,12)(4,13,17,29,16,27,18,28)(9,21,30,32,24,31,25,26)(33,35,38,49,39,47,37,48)(34,40,43,56,44,55,42,45)(36,46,53,62,51,60,52,61)(41,54,50,63,57,64,58,59);(1,4)(2,9)(3,13)(5,17)(6,18)(7,16)(8,21)(10,25)(11,24)(12,26)(14,28)(15,27)(19,29)(20,30)(22,32)(23,31)(33,36,38,53,39,51,37,52)(34,41,43,50,44,57,42,58)(35,46,49,62,47,60,48,61)(40,54,56,63,55,64,45,59);(4,16)(9,24)(13,27)(17,18)(21,31)(25,30)(26,32)(28,29)(35,47)(36,51)(40,55)(41,57)(45,56)(48,49)(50,58)(52,53);(1,5,7,6)(2,10,11,20)(3,14,15,19)(4,17,16,18)(8,22,23,12)(9,25,24,30)(13,28,27,29)(21,32,31,26)(35,47)(40,55)(45,56)(46,60)(48,49)(54,64)(59,63)(61,62)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264654);