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×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1819 |
| commutator | C23×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
662 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 662 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)(3,12,15,7)(4,10,18,8)(5,9)(6,11)(13,20)(14,27,30,21)(16,28,29,23)(17,24,19,25)(22,32,26,31);(1,3,13,29)(2,7,20,28)(4,14,17,30)(5,15,6,16)(8,21,25,27)(9,12,11,23)(10,22,24,26)(18,32,19,31);(2,8,9,24)(4,17)(7,21,12,26)(10,20,25,11)(14,30)(18,19)(22,28,27,23)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=d,e^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 | #72213599, #72235375, #72235377, … |
| maximal normal subgroup of | #72213599, #72235375, #72235377, … |
| maximal quotient of | #72267647 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42732);