Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,5,7,8)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,30,35,31,34,44,50)(27,36,39,46,38,51,29,32)(28,33,43,48,42,49,55,56)(37,52,53,47,40,54,41,45);(1,3,4,7)(2,5,6,8)(11,18)(12,21)(14,17)(15,20)(25,27,31,38)(26,32,34,46)(28,40,42,37)(29,44,39,30)(33,47,49,45)(35,51,50,36)(41,43,53,55)(48,52,56,54);(2,6)(5,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,28)(26,33)(27,37)(29,41)(30,43)(31,42)(32,45)(34,49)(35,48)(36,52)(38,40)(39,53)(44,55)(46,47)(50,56)(51,54);(11,18)(14,17)(19,24)(22,23)(25,29,30,27,31,39,44,38)(26,32,35,36,34,46,50,51)(28,41,43,37,42,53,55,40)(33,45,48,52,49,47,56,54);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(26,34)(27,38)(28,42)(29,39)(35,50)(43,55)(45,47)(52,54)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7092334);