Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12707 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1781 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #10725 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1690 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)(3,11,13,23)(4,15,16,8)(5,18)(6,10)(7,12)(9,19)(14,28,27,24)(17,22,30,29)(20,25)(21,32)(26,31);(1,3,5,14,6,13,19,27)(2,7,9,21,10,20,18,31)(4,12,17,26,16,25,30,32)(8,11,22,24,15,23,29,28);(4,16)(8,15)(11,23)(12,25)(17,30)(22,29)(24,28)(26,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,e^b=ei,f^b=fhi,d^c=di,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240488, #72240494, #72240496, … |
| maximal normal subgroup of | #72240488, #72240494, #72240496, … |
| maximal quotient of | #72281104, C2×#146483 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53679);