Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #271131 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(11,17)(12,19)(14,20)(16,22)(18,23)(21,24)(25,26,30,33)(27,34,37,40)(28,32,29,38)(31,36,39,35);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(27,35)(28,29)(31,34)(32,38)(36,37)(39,40);(1,3,5,4)(2,7,8,6)(9,11,12,17,13,18,19,23)(10,14,16,20,15,21,22,24)(25,27,29,35,30,37,28,36)(26,31,32,34,33,39,38,40);(1,4,5,3)(2,7,8,6)(10,15)(11,18)(16,22)(17,23)(25,28,30,29)(26,32,33,38)(27,36,37,35)(31,34,39,40)Polycyclic presentation:
2^9|a^2=ei,b^2=e,c^2=ef,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 212093);