Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428865 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15391 |
| 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 | |
916 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)(3,14)(4,10)(5,11,25,41)(6,26)(7,12)(8,13)(9,17)(15,34)(16,43,47,39)(18,32)(19,49)(20,51,54,35)(21,37)(22,38)(23,56,48,31)(24,42,27,40)(28,33)(29,36)(30,45)(44,62,52,59)(46,58)(50,57,63,64)(53,60,55,61);(1,3,6,17,8,18,7,28)(2,9,12,14,13,33,26,32)(4,19,21,15,22,45,29,46)(5,23,24,47,25,48,27,16)(10,34,36,49,38,58,37,30)(11,39,40,56,41,43,42,31)(20,52,53,63,54,44,55,50)(35,57,60,62,51,64,61,59);(1,4,8,22)(2,10,13,38)(3,15,18,46)(5,20,25,54)(6,21,7,29)(9,30,33,49)(11,35,41,51)(12,36,26,37)(14,34,32,58)(16,44,47,52)(17,45,28,19)(23,50,48,63)(24,53,27,55)(31,57,56,64)(39,62,43,59)(40,60,42,61);(1,5,6,27,8,25,7,24)(2,11,12,42,13,41,26,40)(3,16,17,48,18,47,28,23)(4,20,21,55,22,54,29,53)(9,31,14,43,33,56,32,39)(10,35,36,61,38,51,37,60)(15,44,45,63,46,52,19,50)(30,57,34,59,49,64,58,62)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336062);