Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #474264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16930 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,13,28,11)(3,14,18,35,19,34,17,8)(4,20,23,9,25,38,24,39)(5,26,29,10,31,41,30,42)(15,32,21,36,46,57,45,56)(16,33,49,59,50,60,48,43)(22,52,27,40,54,37,53,51)(44,61,63,55,64,58,47,62);(1,3)(2,8)(4,21,25,45)(5,16)(6,18)(7,19)(9,36,39,56)(10,33)(11,34)(12,14)(13,35)(15,23,46,24)(17,28)(20,32,38,57)(22,47,54,63)(26,43)(27,44,53,64)(29,49)(30,48)(31,50)(37,61,52,58)(40,62,51,55)(41,59)(42,60);(1,4,6,23,7,25,28,24)(2,9,11,20,13,39,12,38)(3,15,17,45,19,46,18,21)(5,27,29,54,31,53,30,22)(8,32,14,36,35,57,34,56)(10,40,26,52,42,51,41,37)(16,47,48,64,50,63,49,44)(33,58,59,62,60,61,43,55);(1,5,28,29,7,31,6,30)(2,10,12,26,13,42,11,41)(3,16,17,49,19,50,18,48)(4,22,24,27,25,54,23,53)(8,33,14,43,35,60,34,59)(9,37,38,40,39,52,20,51)(15,44,45,63,46,64,21,47)(32,55,36,61,57,62,56,58)Polycyclic presentation:
2^9|a^2=g,b^2=efgh,c^2=e,d^2=fi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,d^c=de,h^c=hi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381461);