Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234321 |
| 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 | |
1191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,36,31,37,29,35)(27,32,40,48,41,49,39,47)(28,34,42,50,43,51,44,33)(38,46,52,55,53,56,54,45);(1,2)(3,8)(4,6)(5,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,31,41)(26,32,37,49)(28,38,43,53)(29,39,30,40)(33,45,50,55)(34,46,51,56)(35,47,36,48)(42,52,44,54);(1,3,4,7)(2,5,6,8)(25,28)(26,33)(27,38)(29,42)(30,44)(31,43)(32,45)(34,35)(36,51)(37,50)(39,52)(40,54)(41,53)(46,47)(48,56)(49,55);(3,7)(5,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,34)(27,38)(29,44)(30,42)(31,43)(32,46)(33,35)(36,50)(37,51)(39,54)(40,52)(41,53)(45,47)(48,55)(49,56);(2,6)(5,8)(10,16)(14,17)(15,24)(22,23)(25,29,31,30)(26,35,37,36)(27,39,41,40)(28,42,43,44)(32,47,49,48)(33,34,50,51)(38,52,53,54)(45,46,55,56)Polycyclic presentation:
2^9|a^2=gh,b^2=fi,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7180825);