Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#121 |
| prime radical | #60558 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,16,14,20,13,19)(11,17,22,23,12,18,21,24);(10,16)(11,12)(13,15)(17,24)(18,23)(19,20);(2,5)(6,8)(9,11,13,21,14,12,15,22)(10,17,19,24,20,18,16,23);(9,12,13,22,14,11,15,21)(10,18,19,23,20,17,16,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=fi,c^2=f,d^2=f,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363622);