Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462138 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16419 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,15,18,35,20,34,19,9)(4,21,24,10,26,38,25,39)(5,27,30,11,31,42,29,41)(16,44,46,59,47,32,22,36)(17,48,50,61,52,33,51,43)(23,54,28,53,56,37,55,40)(45,60,64,58,57,62,49,63);(1,3,7,19,8,20,6,18)(2,9,12,15,14,35,13,34)(4,22,24,47,26,46,25,16)(5,17,30,51,31,52,29,50)(10,36,21,44,39,59,38,32)(11,33,27,43,41,48,42,61)(23,49,28,57,56,64,55,45)(37,62,53,63,54,60,40,58);(1,4,7,24,8,26,6,25)(2,10,12,21,14,39,13,38)(3,16,19,22,20,47,18,46)(5,28,30,56,31,55,29,23)(9,32,15,36,35,44,34,59)(11,40,27,37,41,53,42,54)(17,49,51,57,52,64,50,45)(33,60,43,58,48,62,61,63);(1,5,6,29,8,31,7,30)(2,11,12,27,14,41,13,42)(3,17,18,50,20,52,19,51)(4,23,24,28,26,56,25,55)(9,33,15,43,35,48,34,61)(10,37,38,40,39,54,21,53)(16,45,22,49,47,57,46,64)(32,58,59,60,44,63,36,62)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369334);