Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422611 |
| 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 | |
959 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,14,7,13)(3,15,18,35,20,34,19,9)(4,21,25,39,26,38,24,10)(5,27,29,11,30,41,31,42)(16,32,22,51,45,49,44,36)(17,46,28,57,48,33,47,40)(23,53,55,62,56,37,54,50)(43,60,52,59,63,64,58,61);(1,3,8,20)(2,9,14,35)(4,22)(5,28,30,47)(6,19,7,18)(10,36)(11,40,42,57)(12,34,13,15)(16,25)(17,31,48,29)(21,49)(23,43)(24,45)(26,44)(27,46,41,33)(32,38)(37,61)(39,51)(50,64)(52,54)(53,59)(55,58)(56,63)(60,62);(1,4,6,24,8,26,7,25)(2,10,12,38,14,39,13,21)(3,16,18,44,20,45,19,22)(5,23,29,54,30,56,31,55)(9,32,15,36,35,49,34,51)(11,37,41,62,42,53,27,50)(17,43,28,58,48,63,47,52)(33,59,40,60,46,61,57,64);(1,5,6,29,8,30,7,31)(2,11,12,41,14,42,13,27)(3,17,18,28,20,48,19,47)(4,23,24,54,26,56,25,55)(9,33,15,40,35,46,34,57)(10,37,38,62,39,53,21,50)(16,43,44,58,45,63,22,52)(32,59,36,60,49,61,51,64)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=e,d^2=e,e^2=i,f^2=i,g^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,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329807);