Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17254 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,15,18,34,19,44,31,9)(4,10,22,37,23,38,30,43)(5,24,28,41,29,11,27,42)(16,45,20,32,48,35,47,60)(17,39,50,33,51,55,25,57)(21,52,26,56,53,36,54,40)(46,63,64,62,58,61,49,59);(1,3)(2,9,14,34)(4,20)(5,25)(6,18)(7,31)(8,19)(10,35,38,45)(11,39,24,55)(12,15,13,44)(16,22)(17,27)(21,46)(23,47)(26,49)(28,51)(29,50)(30,48)(32,43,60,37)(33,42,57,41)(36,59,52,62)(40,61,56,63)(53,58)(54,64);(1,4)(2,10)(3,16,19,48)(5,26)(6,30)(7,22)(8,23)(9,32,34,60)(11,40)(12,43)(13,37)(14,38)(15,35,44,45)(17,49,51,64)(18,20,31,47)(21,27)(24,56)(25,58,50,46)(28,53)(29,54)(33,61,57,63)(36,41)(39,62,55,59)(42,52);(1,5,8,29)(2,11,14,24)(3,17,19,51)(4,21,23,53)(6,28,7,27)(9,33,34,57)(10,36,38,52)(12,42,13,41)(15,39,44,55)(16,46,48,58)(18,25,31,50)(20,49,47,64)(22,54,30,26)(32,59,60,62)(35,61,45,63)(37,56,43,40)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386992);