Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401631 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14820 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1735 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,7,12)(3,15,18,9,20,33,19,34)(4,21,24,38,25,37,29,10)(5,26,27,11,28,39,30,40)(16,31,22,35,44,49,43,57)(17,45,46,41,48,32,47,58)(23,52,53,60,55,36,54,50)(42,59,51,63,62,64,61,56);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,24,44,25,43,29,16)(5,17,27,46,28,48,30,47)(10,35,37,31,38,57,21,49)(11,32,26,41,40,45,39,58)(23,51,53,62,55,61,54,42)(36,59,60,56,52,64,50,63);(1,4)(2,10)(3,16)(5,23)(6,29)(7,24)(8,25)(9,31)(11,36)(12,21)(13,37)(14,38)(15,35)(17,42)(18,43)(19,22)(20,44)(26,50)(27,54)(28,55)(30,53)(32,56)(33,57)(34,49)(39,60)(40,52)(41,59)(45,63)(46,61)(47,51)(48,62)(58,64);(1,5,8,28)(2,11,14,40)(3,17,20,48)(4,23,25,55)(6,27,7,30)(9,32,34,45)(10,36,38,52)(12,26,13,39)(15,41,33,58)(16,42,44,62)(18,46,19,47)(21,50,37,60)(22,51,43,61)(24,53,29,54)(31,56,49,63)(35,59,57,64)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,f^c=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, 308828);