Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #435786 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15538 |
| 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 | |
965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,13,6,12)(3,15,20,9,19,33,18,34)(4,21,25,37,26,10,24,38)(5,11,28,40,29,41,30,42)(16,35,22,50,45,48,44,31)(17,39,27,55,46,57,47,32)(23,49,52,60,53,36,54,59)(43,58,61,56,62,63,51,64);(1,3)(2,9)(4,22)(5,27)(6,20)(7,18)(8,19)(10,35)(11,39)(12,33)(13,34)(14,15)(16,25)(17,28)(21,48)(23,43)(24,45)(26,44)(29,47)(30,46)(31,38)(32,40)(36,56)(37,50)(41,57)(42,55)(49,64)(51,52)(53,62)(54,61)(58,59)(60,63);(1,4,8,26)(2,10,13,21)(3,16,19,45)(5,23,29,53)(6,25,7,24)(9,31,34,50)(11,36,41,49)(12,38,14,37)(15,35,33,48)(17,43,46,62)(18,22,20,44)(27,51,47,61)(28,54,30,52)(32,56,55,64)(39,58,57,63)(40,60,42,59);(1,5)(2,11,13,41)(3,17,19,46)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36,21,49)(12,40,14,42)(15,39)(16,43,45,62)(18,27,20,47)(22,51,44,61)(24,52)(25,54)(26,53)(31,56)(33,57)(34,55)(35,58)(37,59,38,60)(48,63)(50,64)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,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, 342983);