Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395242 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×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 | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14,17,15,19,21,18,22)(16,24,26,20,25,23,27,28);(1,2)(3,4)(13,15,17,14,19,22,18,21)(16,23,26,20,25,24,27,28);(2,4)(6,9)(10,12)(14,15,21,22)(16,25)(20,24,28,23)(26,27);(5,7,8,11)(6,10,9,12)(13,16,18,27,19,25,17,26)(14,20,15,23,21,28,22,24)Polycyclic presentation:
2^9|a^2=ef,b^2=h,c^2=e,d^2=fi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 315136);