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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(10,14)(11,17)(12,20)(22,24)(25,26,31,32,30,36,29,35)(27,33,42,44,38,47,39,48)(28,34,43,45,41,49,40,50)(37,46,53,54,52,56,51,55);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,18,17,15)(12,21,20,16)(19,22,23,24)(26,32)(28,40)(29,31)(33,44)(35,36)(37,51)(39,42)(41,43)(45,50)(47,48)(52,53)(54,55);(1,4,5,3)(2,6,7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,27,30,38)(26,33,36,47)(28,37,41,52)(29,39,31,42)(32,44,35,48)(34,46,49,56)(40,51,43,53)(45,54,50,55);(10,14)(11,17)(16,21)(19,23)(25,28,29,40,30,41,31,43)(26,34,35,50,36,49,32,45)(27,37,39,51,38,52,42,53)(33,46,48,55,47,56,44,54);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(27,38)(28,41)(33,47)(34,49)(39,42)(40,43)(44,48)(45,50)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=gi,d^2=fh,e^2=h,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=dh,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, 7107166);