Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501172 |
| 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 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,13)(8,16,17,10)(9,12)(11,15,19,20)(14,18);(1,2)(3,4)(6,10)(7,14)(11,18)(12,16)(13,19)(15,20);(2,4)(8,17)(15,20);(5,7,9,14)(6,11,12,19)(8,15,17,20)(10,13,16,18)Polycyclic presentation:
2^9|a^2=f,d^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,f^b=fh,e^c=ehi,f^d=fh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72263496, #72263497, #72263498, … |
| maximal normal subgroup of | #72263496, #72263497, #72263498, … |
| maximal quotient of | #72284453, #72318159 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408369);