Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458519 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16319 |
| 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 | |
915 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,9,19,31,17,33,18,32)(4,20,22,10,23,38,27,35)(5,24,25,11,26,39,28,37)(15,40,48,56,42,58,43,29)(16,44,49,57,46,59,47,30)(21,34,51,54,52,36,53,50)(41,55,63,61,45,60,62,64);(1,3,7,19,8,17,6,18)(2,9,14,32,12,33,13,31)(4,15,27,48,23,42,22,43)(5,16,28,49,26,46,25,47)(10,29,38,58,35,56,20,40)(11,30,39,59,37,57,24,44)(21,41,53,63,52,45,51,62)(34,55,54,64,36,60,50,61);(1,4,6,22,8,23,7,27)(2,10,13,20,12,35,14,38)(3,15,18,43,17,42,19,48)(5,21,25,51,26,52,28,53)(9,29,31,40,33,56,32,58)(11,36,24,54,37,34,39,50)(16,45,47,63,46,41,49,62)(30,55,44,61,57,60,59,64);(1,5)(2,11)(3,16)(4,21)(6,25)(7,28)(8,26)(9,30)(10,34)(12,37)(13,24)(14,39)(15,41)(17,46)(18,47)(19,49)(20,50)(22,51)(23,52)(27,53)(29,55)(31,44)(32,59)(33,57)(35,36)(38,54)(40,61)(42,45)(43,62)(48,63)(56,60)(58,64)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365716);