Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498904 |
| 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×Q8 | C24 |
| prime radical | #37297 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1323 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1323 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(6,11)(7,9)(8,15)(12,19)(13,16)(17,20);(1,2)(3,4)(6,12)(7,9)(11,19)(13,17)(14,18)(16,20);(1,3,4,2)(5,6)(7,12)(8,16)(9,17)(10,13)(11,15)(14,19)(18,20);(5,7,10,9)(6,12,13,17)(8,14,15,18)(11,20,16,19)Polycyclic presentation:
2^9|c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72260924, #72260939, #72260967, … |
| maximal normal subgroup of | #72260924, #72260939, #72260967, … |
| maximal quotient of | #72261966 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406309);