Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1790 |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #10745 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,15,16,8)(5,9)(6,10)(7,12,19,25)(13,23)(14,24)(17,28,29,21)(18,22)(20,26,30,32)(27,31);(1,3,5,14,6,13,18,27)(2,7,9,20,10,19,22,30)(4,12,17,26,16,25,29,32)(8,11,21,24,15,23,28,31);(8,15)(11,23)(21,28)(24,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,e^b=ehi,f^b=fh,d^c=dh,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240532, #72258095, #72258266, … |
| maximal normal subgroup of | #72240532, #72258095, #72258266, … |
| maximal quotient of | #72280754, C2×#146595, #72285356 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53791);