Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,29,33)(27,35,37,31)(28,39,42,54)(30,44,41,34)(32,47,49,43)(36,51,40,55)(38,46,52,50)(45,53,48,56);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,29,37)(26,31,33,35)(28,40,42,36)(30,38,41,52)(32,48,49,45)(34,46,44,50)(39,51,54,55)(43,56,47,53);(2,5)(6,8)(25,28,41,47,29,42,30,43)(26,32,44,54,33,49,34,39)(27,36,52,56,37,40,38,53)(31,45,50,51,35,48,46,55);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(26,33)(27,37)(28,42)(34,44)(38,52)(43,47)(45,48)(51,55);(2,5)(6,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(28,42)(32,49)(36,40)(39,54)(43,47)(45,48)(51,55)(53,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=hi,e^2=fi,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=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6695191);