Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #289331 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23 | #540 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14)(4,9)(5,20)(6,12)(7,13)(8,16)(10,21)(11,18)(15,29)(17,24)(19,23)(22,27)(25,31)(26,30)(28,32)(33,34,38,42,37,41,44,54)(35,43,47,60,46,39,53,56)(36,48,51,58,50,59,52,40)(45,61,62,55,49,64,63,57);(1,3,6,16,7,4,18,17)(2,8,11,14,13,24,12,9)(5,15,21,30,22,19,32,31)(10,23,20,26,28,29,27,25)(33,35,44,53,37,46,38,47)(34,39,54,60,41,43,42,56)(36,49,52,62,50,45,51,63)(40,57,59,64,58,55,48,61);(1,4)(2,9)(3,7)(5,15)(6,17)(8,11)(10,26)(12,24)(13,14)(16,18)(19,22)(20,29)(21,30)(23,27)(25,28)(31,32)(33,36,37,50)(34,40,41,58)(35,45,46,49)(38,52,44,51)(39,55,43,57)(42,59,54,48)(47,63,53,62)(56,61,60,64);(1,5,18,32,7,22,6,21)(2,10,12,27,13,28,11,20)(3,15,17,31,4,19,16,30)(8,23,9,25,24,29,14,26)(34,41)(39,43)(40,58)(42,54)(48,59)(55,57)(56,60)(61,64)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=ghi,d^c=dh,f^c=fi,g^c=gh53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 196555);