Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432876 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,21)(13,19)(15,23)(16,24)(17,20)(18,22);(3,7)(5,8)(12,22)(16,20)(17,24)(18,21);(2,6)(5,8);(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236543, #72236701, #72237082, … |
| maximal normal subgroup of | #72236543, #72236701, #72237082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340072);