Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4872 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)(10,14)(11,16)(17,18)(19,23)(20,21)(22,24);(6,8)(9,10,13,14)(11,15,16,12)(18,20)(19,24);(5,6)(7,8)(17,19)(18,22)(20,23)(21,24);(1,2)(3,4)(9,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,22)(21,24);(9,12,13,15)(10,16,14,11)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^9|b^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232084, #72232102, #72232136, … |
| maximal normal subgroup of | #72232084, #72232102, #72232136, … |
| maximal quotient of | #72299095, #72299110, #72299341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2712891);