Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×D42 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
8146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6,7,8)(13,14,17,15)(16,19,20,18);(10,12)(14,15)(18,19);(1,2,4,3)(5,6)(7,8)(9,10,11,12);(1,3,4,2)(5,6)(7,8)(13,15,17,14)(16,19,20,18);(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^9|a^2=i,c^2=fghi,d^2=gi,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72232039, #72232043, #72232052, … |
| maximal normal subgroup of | #72232039, #72232043, #72232052, … |
| maximal quotient of | #72252091, #72298107, #72298291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 431731);