Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324436 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(17,18)(19,24)(20,30)(21,26)(22,36)(23,27)(25,32)(28,39)(29,37)(31,42)(33,43)(34,40)(35,46)(38,45)(41,47)(44,48);(2,5)(3,7)(10,14)(11,12)(17,19,23,22)(18,24,27,36)(20,28,33,32)(21,29,34,35)(25,30,39,43)(26,37,40,46)(31,41,44,45)(38,42,47,48);(9,10)(11,16)(12,15)(13,14)(17,20,22,32,23,33,19,28)(18,25,24,30,27,39,36,43)(21,31,35,45,34,44,29,41)(26,38,37,42,40,47,46,48);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,21,23,34)(18,26,27,40)(19,29,22,35)(20,31,33,44)(24,37,36,46)(25,38,39,47)(28,41,32,45)(30,42,43,48);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(21,34)(26,40)(29,35)(31,44)(37,46)(38,47)(41,45)(42,48)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7400047);