Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,36)(28,33)(29,34)(30,44)(31,35)(32,39)(37,50)(38,46)(40,51)(41,47)(42,49)(43,48)(45,52)(53,55)(54,56);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,38,31,40,30,39)(26,32,44,51,35,46,34,36)(28,37,43,54,42,53,41,52)(33,45,47,55,49,56,48,50);(1,3,4,8)(2,6,5,7)(12,20)(15,24)(17,19)(21,23)(25,28,31,42)(26,33,35,49)(27,37,40,53)(29,43,30,41)(32,45,46,56)(34,48,44,47)(36,50,51,55)(38,54,39,52);(10,16)(11,18)(15,24)(17,19)(25,29,31,30)(26,34,35,44)(27,38,40,39)(28,41,42,43)(32,36,46,51)(33,47,49,48)(37,52,53,54)(45,55,56,50)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949854);