Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 | #34935 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,14,17,15,19,13,18)(11,20,12,16,22,23,21,24);(5,6)(7,8)(9,11)(10,16)(12,13)(14,21)(15,22)(17,20)(18,23)(19,24);(1,2,3,4)(6,8)(9,12,13,11,15,21,14,22)(10,16,18,20,19,24,17,23);(6,8)(9,12,13,11,15,21,14,22)(10,16,18,20,19,24,17,23);(9,13,15,14)(10,17,19,18)(11,12,22,21)(16,20,24,23)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72307429, #72307626, #72315247, … |
| maximal normal subgroup of | #72307429, #72307626, #72315247, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351586);