Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6)(7,8)(9,10,15,19)(11,20,13,23)(12,17,14,24)(16,21,18,22);(1,2)(3,4)(5,6)(7,8)(9,11,12,21,15,13,14,22)(10,16,24,23,19,18,17,20);(6,8)(9,12,15,14)(10,17,19,24)(11,21,13,22)(16,20,18,23);(2,4)(6,8)(11,13)(16,18)(20,23)(21,22);(9,13,14,21,15,11,12,22)(10,18,17,23,19,16,24,20)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426227);