Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6660395 |
| 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 ◁ C4×#117 ◁ 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:
(3,7)(6,8)(9,10)(11,18)(12,16)(13,15)(14,17)(19,23)(20,24)(21,22)(25,26,29,35,30,34,41,32)(27,31,38,42,36,44,50,45)(28,33,40,47,39,48,52,46)(37,43,51,53,49,55,56,54);(2,5)(6,8)(9,11,14,13)(10,15,17,18)(12,19,20,21)(16,22,24,23)(25,27,30,36)(26,31,34,44)(28,37,39,49)(29,38,41,50)(32,45,35,42)(33,43,48,55)(40,51,52,56)(46,54,47,53);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,16,17,24)(11,19,13,21)(15,22,18,23)(26,32)(27,36)(28,39)(29,41)(31,42)(33,47)(34,35)(43,54)(44,45)(46,48)(51,56)(53,55);(2,5)(6,8)(10,17)(12,20)(15,18)(19,21)(25,28)(26,33)(27,37)(29,40)(30,39)(31,43)(32,46)(34,48)(35,47)(36,49)(38,51)(41,52)(42,53)(44,55)(45,54)(50,56);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,18,17,15)(12,21,20,19)(16,23,24,22)(26,34)(27,36)(28,39)(32,35)(38,50)(40,52)(43,55)(53,54)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,e^b=ei,d^c=dfh,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, 6574650);