Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2992 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)(9,10,12,16,13,17,14,18)(11,19,21,24,22,23,20,15);(5,6)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19);(1,2,3,4)(6,8)(9,10,14,18,13,17,12,16)(11,19,20,15,22,23,21,24);(6,8)(9,10,12,16,13,17,14,18)(11,19,21,24,22,23,20,15);(9,12,13,14)(10,16,17,18)(11,20,22,21)(15,23,24,19)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72307390, #72307394, #72307626, … |
| maximal normal subgroup of | #72307390, #72307394, #72307626, … |
| maximal quotient of | #72315827 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351549);