Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147773 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10991 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1674 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 1674 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,13,16,27)(4,10,6,8)(5,11)(7,12)(9,15,25,19)(14,30)(17,26,20,23)(18,28)(21,29)(22,31)(24,32);(2,8)(3,14)(4,6)(5,18)(10,12)(11,23)(13,27)(15,31)(16,21)(19,32)(22,24)(26,28);(1,3)(2,9)(4,14)(5,19)(6,21)(7,16)(8,22)(10,24)(11,27)(12,25)(13,28)(15,18)(17,32)(20,31)(23,30)(26,29)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72267961, #72267972, #72267973, … |
| maximal normal subgroup of | #72267961, #72267972, #72267973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54969);