Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422615 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15308 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,29)(3,14,17,34,18,33,28,9)(4,19,22,38,23,37,21,10)(5,24,26,11,27,40,30,41)(15,31,45,50,46,35,44,42)(16,47,25,57,48,32,49,39)(20,52,55,61,56,36,54,51)(43,62,63,58,64,60,53,59);(1,3,8,18)(2,9,13,34)(4,15)(5,25,27,49)(6,28,7,17)(10,35)(11,39,41,57)(12,33,29,14)(16,30,48,26)(19,50)(20,53)(21,45)(22,44)(23,46)(24,47,40,32)(31,38)(36,60)(37,42)(43,54)(51,58)(52,62)(55,64)(56,63)(59,61);(1,4,6,21,8,23,7,22)(2,10,12,37,13,38,29,19)(3,15,17,44,18,46,28,45)(5,20,26,54,27,56,30,55)(9,31,14,42,34,35,33,50)(11,36,40,61,41,52,24,51)(16,43,25,53,48,64,49,63)(32,58,39,62,47,59,57,60);(1,5,6,26,8,27,7,30)(2,11,12,40,13,41,29,24)(3,16,17,25,18,48,28,49)(4,20,21,54,23,56,22,55)(9,32,14,39,34,47,33,57)(10,36,37,61,38,52,19,51)(15,43,44,53,46,64,45,63)(31,58,42,62,35,59,50,60)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 329811);