Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
5410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,22)(20,21)(23,24);(1,2,3,4)(9,11,13,16)(10,14,15,12)(18,21)(22,23);(1,2)(3,4)(9,12)(10,11)(13,14)(15,16)(17,19,20,24)(18,22,21,23);(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,24)(21,22)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72230894, #72231761, #72232313, … |
| maximal normal subgroup of | #72230894, #72231761, #72232313, … |
| maximal quotient of | #72264980, #72266798, #72278103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4029851);