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 | |
993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,7,12,6,11)(3,13,16,29,15,28,17,30)(4,18,20,27,19,9,21,25)(8,22,23,14,24,31,26,32);(1,3,6,17,7,15,5,16)(2,8,11,23,12,24,10,26)(4,14,20,31,19,32,21,22)(9,13,25,30,18,28,27,29);(9,18)(13,28)(25,27)(29,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=i,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 | #72317371, #72317378, #72317390, … |
| maximal normal subgroup of | #72317371, #72317378, #72317390, … |
| maximal quotient of | C2×#146598 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53794);