Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #917169 |
| automorphism | ? |
| 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 | |
3650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,23,24,21);(6,8)(9,11,14,16)(10,15,12,13)(17,18)(19,21)(20,22)(23,24);(5,6,7,8)(10,12)(11,16)(17,18,20,22)(19,21,24,23);(1,2)(3,4)(9,12,14,10)(11,15,16,13)(17,18)(19,21)(20,22)(23,24);(9,13,14,15)(10,16,12,11)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72297354, #72297366, #72297403, … |
| maximal normal subgroup of | #72297354, #72297366, #72297403, … |
| maximal quotient of | #72298237, #72298254, #72298296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 825970);