Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459921 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16352 |
| 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 | |
893 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,6,13,7,27)(3,14,17,33,18,9,28,39)(4,19,29,35,21,48,22,10)(5,23,30,51,25,11,26,36)(15,40,42,37,43,46,55,31)(16,32,44,60,45,38,56,59)(20,47,54,50,24,52,49,34)(41,61,62,58,53,63,64,57);(1,3,8,28,6,18,7,17)(2,9,27,39,13,14,12,33)(4,15,29,55,21,43,22,42)(5,16,30,56,25,45,26,44)(10,31,48,40,35,37,19,46)(11,32,51,60,23,38,36,59)(20,41,54,64,24,53,49,62)(34,57,52,61,50,58,47,63);(1,4,7,22,6,21,8,29)(2,10,12,19,13,35,27,48)(3,15,17,42,18,43,28,55)(5,24,26,54,25,20,30,49)(9,31,33,46,14,37,39,40)(11,34,36,47,23,50,51,52)(16,41,44,62,45,53,56,64)(32,58,59,61,38,57,60,63);(1,5,7,26,6,25,8,30)(2,11,12,36,13,23,27,51)(3,16,17,44,18,45,28,56)(4,20,22,49,21,24,29,54)(9,32,33,59,14,38,39,60)(10,34,19,47,35,50,48,52)(15,41,42,62,43,53,55,64)(31,57,46,63,37,58,40,61)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=f,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367118);