Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #402444 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14842 |
| 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 | |
934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,14,6,13)(3,15,19,32,20,39,18,9)(4,21,23,36,24,35,28,10)(5,25,29,38,27,11,26,37)(16,33,43,40,45,30,44,59)(17,46,48,31,49,41,47,54)(22,51,52,34,53,50,56,55)(42,57,61,64,63,58,62,60);(1,3,6,18,7,20,8,19)(2,9,12,15,14,32,13,39)(4,16,23,43,24,45,28,44)(5,17,26,47,27,49,29,48)(10,33,35,59,36,30,21,40)(11,31,37,41,25,54,38,46)(22,42,52,61,53,63,56,62)(34,58,51,64,55,57,50,60);(1,4,7,24)(2,10)(3,16)(5,22,27,53)(6,23,8,28)(9,30,32,33)(11,34)(12,35)(13,21)(14,36)(15,40,39,59)(17,42)(18,43)(19,44)(20,45)(25,55)(26,52,29,56)(31,57,54,58)(37,51)(38,50)(41,60,46,64)(47,61)(48,62)(49,63);(1,5,6,26,7,27,8,29)(2,11,12,37,14,25,13,38)(3,17,18,47,20,49,19,48)(4,22,23,52,24,53,28,56)(9,31,15,41,32,54,39,46)(10,34,35,51,36,55,21,50)(16,42,43,61,45,63,44,62)(30,57,40,60,33,58,59,64)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309640);