Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498886 |
| 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 | #37269 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(6,10)(7,13)(8,15)(11,18)(12,16)(17,19);(1,2)(3,4)(8,15)(14,20);(1,3,4,2)(5,6)(7,11)(8,16)(9,12)(10,15)(13,19)(14,18)(17,20);(5,7,9,13)(6,11,12,19)(8,14,15,20)(10,17,16,18)Polycyclic presentation:
2^9|c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,f^c=fi,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72260921, #72260939, #72260953, … |
| maximal normal subgroup of | #72260921, #72260939, #72260953, … |
| maximal quotient of | #72261831, C2×#499139 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406335);