Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233936 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#2537 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(6,9)(7,11)(10,12)(13,14,16,17)(15,18,20,19);(1,2)(3,4)(13,15)(14,18)(16,20)(17,19);(1,3,4,2)(5,6)(7,12)(8,10)(9,11)(14,17)(15,19)(18,20);(6,10)(9,12)Polycyclic presentation:
2^9|a^2=i,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=cg,g^b=gi,d^c=dh,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72264035, #72264046, #72264054, … |
| maximal normal subgroup of | #72264035, #72264046, #72264054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145872);