Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417778 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1927 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(6,8)(9,10,14,19)(11,20,25,36)(12,16,29,35)(13,17,21,18)(15,23,31,24)(22,37,39,32)(26,40,38,30)(27,34,28,33);(1,2,3,4)(9,11,21,23,14,25,13,24)(10,15,17,36,19,31,18,20)(12,26,28,22,29,38,27,39)(16,32,34,30,35,37,33,40);(5,6,7,8)(9,12)(10,16)(11,22)(13,28)(14,29)(15,30)(17,33)(18,34)(19,35)(20,37)(21,27)(23,26)(24,38)(25,39)(31,40)(32,36);(2,4)(10,17)(11,23)(13,21)(15,31)(16,33)(18,19)(22,26)(24,25)(27,28)(30,40)(34,35)(38,39)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330073);