Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15323 |
| 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 | |
1182 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)(3,14,17,9)(4,10)(5,25)(6,13)(7,29)(8,12)(11,27)(15,30,43,41)(16,45,46,56)(18,39,19,32)(20,33,44,59)(21,50)(22,36)(23,35)(24,55)(26,38)(28,40)(31,48,37,47)(34,54)(42,63,64,58)(49,57,51,62)(52,61)(53,60);(1,3)(2,9)(4,20)(5,16)(6,19)(7,18)(8,17)(10,33)(11,37)(12,14)(13,39)(15,23)(21,51)(22,44)(24,43)(25,56)(26,48)(27,47)(28,46)(29,32)(30,35)(31,38)(34,58)(36,59)(40,45)(41,55)(42,53)(49,52)(50,62)(54,64)(57,61)(60,63);(1,4)(2,10)(3,15,17,43)(5,21)(6,23)(7,24)(8,22)(9,30,14,41)(11,34)(12,36)(13,35)(16,42,46,64)(18,20,19,44)(25,50)(26,53)(27,54)(28,52)(29,55)(31,57,37,62)(32,33,39,59)(38,60)(40,61)(45,58,56,63)(47,51,48,49);(1,5,8,28)(2,11,12,38)(3,16,17,46)(4,21,22,52)(6,27,7,26)(9,31,14,37)(10,34,36,60)(13,40,29,25)(15,42,43,64)(18,48,19,47)(20,49,44,51)(23,54,24,53)(30,57,41,62)(32,45,39,56)(33,58,59,63)(35,61,55,50)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^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,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331810);