Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,8)(7,11,12,10)(13,14,17,22,19,24,18,23)(15,20,25,28,21,16,26,27);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,25,19,21,18,26)(14,20,22,28,24,16,23,27);(2,4)(5,8,9,6)(7,11,12,10)(13,16,19,20)(14,21,24,15)(17,28,18,27)(22,25,23,26);(7,12)(10,11)(13,14)(15,20)(16,21)(17,22)(18,23)(19,24)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^c=fi,g^c=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329090);