Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1819 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
628 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 628 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,5,12,7,13,3,14)(4,11,16,30,20,9,21,28)(6,10,22,26,15,23,17,8)(18,27,31,24,19,29,32,25);(1,3,15,22)(2,8,23,12)(4,16,18,31)(5,6,17,7)(9,24,29,30)(10,14,13,26)(11,25,27,28)(19,32,20,21);(2,9,10,27)(4,18)(8,24,14,28)(11,23,29,13)(12,30,26,25)(16,31)(19,20)(21,32)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=egi,f^c=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72235018, #72235020, #72260536, … |
| maximal normal subgroup of | #72235018, #72235020, #72260536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42736);