Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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 |
| prime radical | #35223 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3337 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)(9,10,14,16)(11,17,18,24)(12,13,20,21)(15,22,23,19);(5,6)(7,8)(9,11,13,19,14,18,21,22)(10,15,12,17,16,23,20,24);(2,4)(6,8)(9,12,13,16,14,20,21,10)(11,17,19,23,18,24,22,15);(6,8);(11,18)(15,23)(17,24)(19,22)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72313028, #72313036, #72313076 |
| maximal normal subgroup of | #72313028, #72313036, #72313076 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346140);