Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417460 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,12)(7,10,8,11)(13,14,17,15)(16,21,24,19)(18,22,26,20)(23,25,28,27);(5,7,9,8)(6,10,12,11)(13,15,17,14)(16,23)(18,22,26,20)(19,25)(21,27)(24,28);(5,7,9,8)(6,11,12,10)(13,16,17,24)(14,19,15,21)(18,27,26,25)(20,23,22,28);(2,4)(5,8,9,7)(6,11,12,10)(18,26)(20,22)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339377);