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) = C28×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 | |
5234 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)(11,13)(12,15)(14,16)(17,18,22,21)(19,23,20,24);(6,8)(10,11)(12,16)(18,21)(23,24);(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,19,22,20)(18,23,21,24);(1,2)(3,4)(9,11,13,10)(12,14,16,15)(17,20,22,19)(18,23,21,24);(9,12,13,16)(10,14,11,15)(17,21,22,18)(19,24,20,23)Polycyclic presentation:
2^9|a^2=fg,c^2=f,d^2=f,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 | #72232595, #72232623, #72232659, … |
| maximal normal subgroup of | #72232595, #72232623, #72232659, … |
| maximal quotient of | #72278203, #72278213, #72278645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2712870);