Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462891 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16454 |
| 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 | |
649 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)(3,15,19,34)(4,20)(5,26)(6,12)(7,13)(8,14)(9,18,35,31)(10,23)(11,30)(16,44,46,32)(17,47,50,43)(21,36,51,60)(22,40)(24,39)(25,38)(27,56)(28,42)(29,41)(33,58,55,49)(37,53)(45,59,57,63)(48,61,64,62)(52,54);(1,3,8,19)(2,9)(4,21,25,51)(5,17,30,50)(6,18,7,31)(10,36)(11,33)(12,15)(13,34)(14,35)(16,23,46,24)(20,44)(22,48,54,64)(26,55)(27,57,53,45)(28,49,29,58)(32,38)(37,62)(39,60)(40,59)(41,43)(42,47)(52,63)(56,61);(1,4,8,25)(2,10,14,39)(3,16,19,46)(5,27,30,53)(6,24,7,23)(9,32,35,44)(11,40,26,52)(12,20,13,38)(15,36,34,60)(17,48,50,64)(18,51,31,21)(22,29,54,28)(33,61,55,62)(37,42,56,41)(43,59,47,63)(45,58,57,49);(1,5,6,28,8,30,7,29)(2,11,13,42,14,26,12,41)(3,17,18,49,19,50,31,58)(4,22,23,27,25,54,24,53)(9,33,34,47,35,55,15,43)(10,37,20,52,39,56,38,40)(16,45,21,48,46,57,51,64)(32,59,36,62,44,63,60,61)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=i,d^2=f,e^2=i,f^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,f^c=fi,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, 370088);