Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #371873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14082 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,26)(3,9,17,32,16,33,27,31)(4,18,23,49,21,10,22,36)(5,11,25,39,24,37,28,38)(14,40,19,34,42,29,51,47)(15,43,46,57,44,30,45,56)(20,35,52,60,53,48,54,59)(41,61,63,64,62,55,50,58);(1,3,6,17,8,16,7,27)(2,9,13,31,12,33,26,32)(4,19,23,14,21,51,22,42)(5,15,25,45,24,44,28,46)(10,34,49,29,18,47,36,40)(11,30,38,56,37,43,39,57)(20,50,54,41,53,63,52,62)(35,58,60,55,48,64,59,61);(1,4)(2,10,12,18)(3,14,16,42)(5,20)(6,22)(7,23)(8,21)(9,29)(11,35,37,48)(13,36,26,49)(15,41,44,62)(17,19,27,51)(24,53)(25,52)(28,54)(30,55)(31,34)(32,47)(33,40)(38,59,39,60)(43,61)(45,50,46,63)(56,58)(57,64);(1,5,6,25,8,24,7,28)(2,11,13,38,12,37,26,39)(3,15,17,45,16,44,27,46)(4,20,22,52,21,53,23,54)(9,30,31,56,33,43,32,57)(10,35,36,59,18,48,49,60)(14,41,19,50,42,62,51,63)(29,55,34,58,40,61,47,64)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=ef,d^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279072);