Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,14,17)(11,18,19,24)(12,16,13,22)(15,20,23,21);(5,6)(7,8)(9,11,13,21,14,19,12,20)(10,15,16,18,17,23,22,24);(2,4)(6,8)(11,19)(15,23)(18,24)(20,21);(5,6)(7,8)(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,18,23,24);(6,8)(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,18,23,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481241);