Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,7,8)(9,10)(11,12)(14,18)(15,16)(19,20);(1,2,3,4)(13,14,15,18,17,20,16,19);(6,8)(14,18)(15,16)(19,20);(2,4)(10,12)(14,19)(15,16)(18,20)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239711, #72239726, #72239733, … |
| maximal normal subgroup of | #72239711, #72239726, #72239733, … |
| maximal quotient of | #72250225, #72257228, #72320443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326146);