Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396717 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14737 |
| 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 | |
1113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,18,9)(4,21,26,37)(5,27,29,40)(6,13,7,12)(10,25,38,24)(11,30,39,28)(16,43,45,35)(17,47,48,56)(19,33,20,34)(22,41,46,31)(23,36,55,60)(32,50,42,49)(44,58,63,61)(51,53,57,54)(52,62,64,59);(1,3,6,20,8,18,7,19)(2,9,12,34,14,15,13,33)(4,22,24,16,26,46,25,45)(5,17,28,50,29,48,30,49)(10,35,21,31,38,43,37,41)(11,32,27,56,39,42,40,47)(23,52,53,44,55,64,54,63)(36,59,51,58,60,62,57,61);(1,4)(2,10)(3,16,18,45)(5,23)(6,25)(7,24)(8,26)(9,31,15,41)(11,36)(12,37)(13,21)(14,38)(17,44,48,63)(19,46,20,22)(27,57)(28,54)(29,55)(30,53)(32,58,42,61)(33,43,34,35)(39,60)(40,51)(47,62,56,59)(49,64,50,52);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,27)(13,40)(14,39)(15,42)(16,44)(18,48)(19,49)(20,50)(21,51)(22,52)(24,53)(25,54)(26,55)(31,58)(33,47)(34,56)(35,59)(37,57)(38,60)(41,61)(43,62)(45,63)(46,64)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303914);