Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7207830 |
| 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 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1196 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)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,30,35)(27,36,31,32)(28,39,41,46)(29,34,45,52)(33,37,49,42)(38,50,43,47)(40,55,44,56)(48,53,51,54);(1,2)(3,8)(4,5)(6,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27,30,31)(26,32,35,36)(28,37,41,42)(29,43,45,38)(33,46,49,39)(34,50,52,47)(40,54,44,53)(48,55,51,56);(2,5)(6,8)(25,28,31,42,30,41,27,37)(26,33,36,39,35,49,32,46)(29,44,43,53,45,40,38,54)(34,51,50,56,52,48,47,55);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(22,23)(25,29)(26,34)(27,38)(28,40)(30,45)(31,43)(32,47)(33,48)(35,52)(36,50)(37,53)(39,55)(41,44)(42,54)(46,56)(49,51);(2,5)(6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(26,35)(29,45)(32,36)(33,49)(38,43)(39,46)(40,44)(53,54)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7132175);