Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4664 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)(9,10,13,15)(11,12,16,14)(17,18)(19,22)(20,21)(23,24);(6,8)(9,11,13,16)(10,14,15,12)(18,21)(19,23);(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19,20,23)(18,22,21,24);(1,2)(3,4)(9,12)(10,11)(13,14)(15,16);(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280833, #72280898, #72281213, … |
| maximal normal subgroup of | #72280833, #72280898, #72281213, … |
| maximal quotient of | #72323510, C2×#647764, #72330602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 554960);