Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405946 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14921 |
| 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 | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,6,13,7,25)(3,9,26,31,16,32,17,37)(4,18,28,34,21,47,22,10)(5,11,27,35,23,36,24,38)(14,39,49,55,19,45,41,29)(15,30,44,56,42,57,43,58)(20,50,53,59,51,46,52,33)(40,61,63,64,48,60,62,54);(1,3)(2,9)(4,19,21,14)(5,15)(6,16)(7,26)(8,17)(10,29,34,55)(11,30)(12,37)(13,32)(18,45,47,39)(20,48,51,40)(22,41,28,49)(23,42)(24,44)(25,31)(27,43)(33,54,59,64)(35,58)(36,57)(38,56)(46,61,50,60)(52,62,53,63);(1,4,6,21)(2,10)(3,14,16,19)(5,20,23,51)(7,22,8,28)(9,29)(11,33)(12,18)(13,34)(15,40,42,48)(17,41,26,49)(24,52,27,53)(25,47)(30,54)(31,39)(32,55)(35,50)(36,59)(37,45)(38,46)(43,62,44,63)(56,61)(57,64)(58,60);(1,5,6,23)(2,11,13,36)(3,15,16,42)(4,20,21,51)(7,27,8,24)(9,30,32,57)(10,33,34,59)(12,38,25,35)(14,40,19,48)(17,44,26,43)(18,46,47,50)(22,53,28,52)(29,54,55,64)(31,58,37,56)(39,60,45,61)(41,63,49,62)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313143);