Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| 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 | |
3078 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)(9,10,12,16,14,18,13,17)(11,19,21,24,22,23,20,15);(5,6)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24);(1,2,3,4)(6,8)(9,10,12,16,14,18,13,17)(11,19,21,24,22,23,20,15);(6,8)(9,10,12,16,14,18,13,17)(11,19,21,24,22,23,20,15);(9,12,14,13)(10,16,18,17)(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=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72307380, #72307404, #72307626, … |
| maximal normal subgroup of | #72307380, #72307404, #72307626, … |
| maximal quotient of | #72315743 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351533);