Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436027 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15546 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,28,7,13,6,12)(3,14,27,9,18,38,17,33)(4,19,21,48,22,10,29,35)(5,23,30,37,26,11,25,53)(15,40,42,31,43,39,52,46)(16,44,54,57,45,36,24,32)(20,34,50,59,51,47,55,63)(41,58,49,62,61,60,64,56);(1,3,7,18)(2,9,13,33)(4,15,22,43)(5,24,26,54)(6,27,8,17)(10,31,19,46)(11,36,23,44)(12,38,28,14)(16,30,45,25)(20,49,51,64)(21,52,29,42)(32,53,57,37)(34,58,47,60)(35,40,48,39)(41,55,61,50)(56,63,62,59);(1,4,6,21,7,22,8,29)(2,10,12,35,13,19,28,48)(3,15,17,42,18,43,27,52)(5,20,25,50,26,51,30,55)(9,31,14,39,33,46,38,40)(11,34,37,59,23,47,53,63)(16,41,24,49,45,61,54,64)(32,56,36,58,57,62,44,60);(1,5,6,25,7,26,8,30)(2,11,28,53,13,23,12,37)(3,16,17,24,18,45,27,54)(4,20,21,50,22,51,29,55)(9,32,38,44,33,57,14,36)(10,34,48,63,19,47,35,59)(15,41,42,49,43,61,52,64)(31,56,40,60,46,62,39,58)Polycyclic presentation:
2^9|a^2=eg,b^2=i,c^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343224);