Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| 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 | |
1162 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)(6,10)(11,12)(13,14)(15,20)(16,17)(18,21)(19,22)(23,24)(25,28)(26,27);(5,6,9,10)(7,11,8,12)(13,15,18,26,19,25,17,24)(14,20,21,27,22,28,16,23);(5,7,9,8)(6,11,10,12)(13,16,17,14,19,21,18,22)(15,23,26,28,25,27,24,20);(2,4)(5,8,9,7)(6,11,10,12)(13,17,19,18)(14,21,22,16)(15,24,25,26)(20,27,28,23)Polycyclic presentation:
2^9|b^2=efg,c^2=efi,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346520);