Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6443048 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,18,14,19,15,24)(11,20,17,12,22,16,21,23);(1,2)(3,7)(4,5)(6,8)(9,11,15,17,14,22,13,21)(10,16,18,12,19,20,24,23);(2,5)(6,7)(9,12,15,20,14,23,13,16)(10,17,18,22,19,21,24,11);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,18,19,24)(11,17,22,21)(12,16,23,20);(9,13,14,15)(10,18,19,24)(11,21,22,17)(12,20,23,16)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=g,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371293);