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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1349 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)(3,7,8,5)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,29,34,30,35,41,31)(27,32,38,47,37,46,52,43)(28,33,39,50,42,49,54,44)(36,45,51,56,40,48,53,55);(1,3,4,8)(2,5,6,7)(11,17)(12,20)(14,21)(15,19)(26,31)(28,39)(29,41)(32,43)(34,35)(36,51)(38,52)(40,53)(42,54)(44,50)(46,47)(55,56);(2,6)(5,7)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,27)(26,32)(28,40)(29,38)(30,37)(31,43)(33,48)(34,47)(35,46)(36,42)(39,53)(41,52)(44,56)(45,49)(50,55)(51,54);(3,8)(5,7)(11,17)(14,21)(18,23)(22,24)(25,28,41,54,30,42,29,39)(26,33,31,44,35,49,34,50)(27,36,52,53,37,40,38,51)(32,45,43,55,46,48,47,56);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,37)(28,42)(32,46)(33,49)(38,52)(39,54)(43,47)(44,50)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=h,d^2=fgi,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=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, 7105149);