Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454426 |
| 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 | |
1072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,17,21)(15,16,24,19)(18,22,27,23)(20,25,28,26);(6,9)(10,12)(13,15,18,20,17,24,27,28)(14,19,22,26,21,16,23,25);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16,17,19)(14,15,21,24)(18,26,27,25)(20,22,28,23);(7,11)(10,12)(13,16)(14,20)(15,23)(17,19)(18,25)(21,28)(22,24)(26,27)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372200);