Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395986 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14729 |
| 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 | |
1287 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)(4,21)(5,27,29,39)(6,13)(7,12)(8,14)(9,18)(10,25)(11,28,40,30)(16,31)(17,32,48,57)(19,34)(20,33)(22,35)(23,36,54,60)(24,38)(26,37)(41,44)(42,46,55,47)(43,61,63,58)(45,49)(50,53,59,52)(51,56,62,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,24,45,26,44,25,16)(5,17,28,46,29,48,30,47)(10,35,37,31,38,41,21,49)(11,32,39,42,40,57,27,55)(23,51,52,63,54,62,53,43)(36,58,59,56,60,61,50,64);(1,4,7,25,8,26,6,24)(2,10,13,21,14,38,12,37)(3,16,19,44,20,45,18,22)(5,23,30,53,29,54,28,52)(9,31,33,35,34,49,15,41)(11,36,27,50,40,60,39,59)(17,43,47,62,48,63,46,51)(32,56,55,58,57,64,42,61);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,39)(13,27)(14,40)(15,42)(16,43)(18,46)(19,47)(20,48)(21,50)(22,51)(24,52)(25,53)(26,54)(31,56)(33,55)(34,57)(35,58)(37,59)(38,60)(41,61)(44,62)(45,63)(49,64)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=ei,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,g^b=gi,g^c=gi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303183);