Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×D42 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,21)(22,24);(2,4)(6,8)(10,13)(14,16)(18,21)(22,23);(1,2)(3,4)(5,6)(7,8)(11,15)(14,16)(17,19,20,24)(18,22,21,23);(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23);(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|c^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72236761, #72237108, #72237111, … |
| maximal normal subgroup of | #72236761, #72237108, #72237111, … |
| maximal quotient of | #72250288, #72293977, C2×#703992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 611188);