Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511779 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #24260 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20);(1,2,3,4)(7,14)(10,17)(13,19)(15,20);(5,7,9,14)(6,10,12,17)(8,15,16,20)(11,13,18,19);(8,16)(15,20)Polycyclic presentation:
2^9|b^2=e,c^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,g^c=ghi,f^d=fh,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72281473, #72281494, #72281755, … |
| maximal normal subgroup of | #72281473, #72281494, #72281755, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419136);