Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500450 |
| 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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,34)(27,35,37,49)(28,32,39,46)(29,41,42,33)(31,43,45,38)(36,48,51,56)(40,53,52,47)(44,54,55,50);(2,5)(6,8)(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)(25,27,29,38,30,37,42,43)(26,31,33,35,34,45,41,49)(28,36,40,50,39,51,52,54)(32,44,47,48,46,55,53,56);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(25,28,30,39)(26,32,34,46)(27,36,37,51)(29,40,42,52)(31,44,45,55)(33,47,41,53)(35,48,49,56)(38,50,43,54);(11,18)(14,22)(17,19)(21,23)(27,37)(28,39)(31,45)(32,46)(35,49)(38,43)(40,52)(47,53)Polycyclic presentation:
2^9|a^2=hi,c^2=g,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6418156);