Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1790 |
| commutator | C2×Q8 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #10745 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
970 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)(3,14,18,29)(4,11,5,21)(6,12)(7,23)(8,13)(9,16)(10,20,28,22)(15,27)(17,24,19,32)(25,31)(26,30);(1,3,7,19,8,18,6,17)(2,9,12,26,13,27,23,25)(4,15,20,30,5,16,22,31)(10,24,21,29,28,32,11,14);(1,4)(2,10)(3,15)(5,8)(6,22)(7,20)(9,24)(11,23)(12,21)(13,28)(14,25)(16,18)(17,31)(19,30)(26,29)(27,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,e^b=eh,f^b=fhi,d^c=dhi,e^c=ei,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317333, #72317375, #72317465, … |
| maximal normal subgroup of | #72317333, #72317375, #72317465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53843);