Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #289395 |
| 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 | #532 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,10,4,16)(5,19,21,32)(6,23,8,12)(9,18,25,15)(11,22,29,20)(14,27,17,24)(26,31,28,30)(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);(1,3,8,18,7,4,6,15)(2,9,23,16,13,25,12,10)(5,17,22,30,21,14,20,31)(11,27,32,26,29,24,19,28)(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)(2,10)(3,7)(5,14)(6,18)(8,15)(9,23)(11,28)(12,25)(13,16)(17,21)(19,24)(20,30)(22,31)(26,29)(27,32)(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);(1,5,8,22,7,21,6,20)(2,11,23,32,13,29,12,19)(3,14,18,31,4,17,15,30)(9,24,16,28,25,27,10,26)(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,d^b=dh,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, 196627);