Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1850 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)(3,5)(4,6)(7,8)(9,10,15,18,16,12,14,19)(11,20,21,17,23,24,22,13);(1,2)(3,7)(4,6)(5,8)(9,11,15,21,16,23,14,22)(10,17,19,20,12,13,18,24);(1,3)(2,5)(4,8)(6,7)(9,12,16,10)(11,17,23,13)(14,19,15,18)(20,22,24,21);(3,8)(5,7)(9,13)(10,11)(12,23)(14,24)(15,20)(16,17)(18,21)(19,22);(2,6)(5,7)(9,14,16,15)(10,18,12,19)(11,21,23,22)(13,24,17,20)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910273);