Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,22)(18,24)(21,23)(25,26)(27,35)(28,39)(29,43)(30,33)(31,38)(32,41)(34,44)(36,47)(37,50)(40,45)(42,55)(46,53)(48,51)(49,52)(54,56);(2,5)(6,8)(9,11,13,17)(10,14,15,22)(12,18,20,23)(16,21,19,24)(25,27,29,38,30,37,44,53)(26,31,34,35,33,46,43,50)(28,40,41,56,42,36,51,52)(32,47,39,54,48,45,55,49);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,19)(18,23)(25,28,29,41,30,42,44,51)(26,32,34,39,33,48,43,55)(27,36,38,52,37,40,53,56)(31,45,35,49,46,47,50,54);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,21,22,24)(26,33)(31,46)(32,48)(34,43)(35,50)(39,55)(45,47)(49,54);(27,37)(31,46)(35,50)(36,40)(38,53)(45,47)(49,54)(52,56)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=f,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343539);