Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146581 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10735 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1687 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 1687 conjugacy classes of subgroups — too many to draw. Only its 200 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11,13,22)(4,8,15,23)(5,17,19,9)(6,10)(7,12)(14,26)(16,25)(18,24)(20,27)(21,30,31,28)(29,32);(1,3,5,14,6,13,19,29)(2,7,9,21,10,20,17,31)(4,12,16,28,15,27,18,30)(8,22,24,32,23,11,25,26);(4,15)(8,23)(11,22)(12,27)(16,18)(24,25)(26,32)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,h^b=hi,d^c=di,f^d=fi,g^d=gi,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213476, #72213538, #72213555, … |
| maximal normal subgroup of | #72213476, #72213538, #72213555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53778);