Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462962 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16458 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,13,8,29)(3,14,30,9,18,33,17,34)(4,10,21,37,22,38,23,41)(5,24,26,55,27,11,28,40)(15,35,49,60,44,59,19,31)(16,45,57,53,48,42,47,32)(20,50,25,54,51,36,52,39)(43,61,64,58,56,62,46,63);(1,3,7,18)(2,9)(4,19,22,49)(5,16,27,48)(6,17,8,30)(10,35)(11,32)(12,14)(13,34)(15,21,44,23)(20,46,51,64)(24,53)(25,56,52,43)(26,47,28,57)(29,33)(31,37)(36,62)(38,59)(39,58)(40,42)(41,60)(45,55)(50,61)(54,63);(1,4,7,22)(2,10,13,38)(3,15,18,44)(5,25,27,52)(6,23,8,21)(9,31,34,60)(11,39,24,54)(12,41,29,37)(14,35,33,59)(16,46,48,64)(17,49,30,19)(20,28,51,26)(32,61,53,62)(36,55,50,40)(42,58,45,63)(43,57,56,47);(1,5,8,28,7,27,6,26)(2,11,12,40,13,24,29,55)(3,16,30,57,18,48,17,47)(4,20,23,52,22,51,21,25)(9,32,14,42,34,53,33,45)(10,36,37,39,38,50,41,54)(15,43,49,64,44,56,19,46)(31,58,59,61,60,63,35,62)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=i,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,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, 370159);