Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,34)(27,31,36,45)(28,38,40,55)(29,42,43,33)(32,47,49,41)(35,50,39,54)(37,53,52,46)(44,56,48,51);(3,4)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27)(26,31)(28,39)(29,37)(30,36)(32,48)(33,46)(34,45)(35,40)(38,54)(41,56)(42,53)(43,52)(44,49)(47,51)(50,55);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,29,41,30,40,43,47)(26,32,33,38,34,49,42,55)(27,35,37,51,36,39,52,56)(31,44,46,50,45,48,53,54);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(27,36)(28,40)(31,45)(32,49)(37,52)(38,55)(41,47)(46,53);(1,4,5,3)(2,6,7,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,36)(31,45)(35,39)(37,52)(44,48)(46,53)(50,54)(51,56)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6705644);