Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1781 |
| prime radical | #62811 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
3386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13,16,26)(4,10)(5,18,20,8)(6,11)(7,12)(9,14,25,21)(15,29)(17,24,19,23)(22,28)(27,30)(31,32);(2,8)(9,22)(10,23)(11,24)(12,18)(13,27)(25,31)(26,29);(1,3)(2,9)(4,14)(5,15)(6,21)(7,16)(8,22)(10,13)(11,26)(12,25)(17,28)(18,31)(19,32)(20,30)(23,27)(24,29);(1,4,7,6)(2,10,12,11)(3,14,16,21)(5,17,20,19)(8,23,18,24)(9,13,25,26)(15,28,30,32)(22,27,31,29)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gh,f^c=fi,g^c=gi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237529, #72237540, #72237542, … |
| maximal normal subgroup of | #72237529, #72237540, #72237542, … |
| maximal quotient of | #72260371, #72260453, #72267978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400606);