Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #371877 |
| 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 | |
1027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,27)(3,9,17,32,16,33,28,31)(4,18,22,10,21,36,23,37)(5,11,26,38,24,40,25,39)(14,41,19,50,44,34,43,29)(15,45,46,57,48,30,47,58)(20,52,53,35,54,49,55,60)(42,61,63,56,62,59,51,64);(1,3,6,17,8,16,7,28)(2,9,13,31,12,33,27,32)(4,19,22,14,21,43,23,44)(5,15,26,47,24,48,25,46)(10,34,18,29,37,41,36,50)(11,30,39,58,40,45,38,57)(20,51,55,42,54,63,53,62)(35,59,49,56,60,61,52,64);(1,4,6,23,8,21,7,22)(2,10,27,18,12,37,13,36)(3,14,28,43,16,44,17,19)(5,20,26,53,24,54,25,55)(9,29,31,34,33,50,32,41)(11,35,38,49,40,60,39,52)(15,42,46,63,48,62,47,51)(30,56,58,59,45,64,57,61);(1,5,6,26,8,24,7,25)(2,11,13,39,12,40,27,38)(3,15,17,47,16,48,28,46)(4,20,23,53,21,54,22,55)(9,30,31,58,33,45,32,57)(10,35,36,52,37,60,18,49)(14,42,19,51,44,62,43,63)(29,56,34,59,50,64,41,61)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=ef,d^2=g,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,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, 279076);