Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#99 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×C4×#99 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,4,9,18)(3,12,15,28,16,27,22,32)(5,17,19,31,20,10,24,25)(8,21,23,30,13,26,29,14);(1,3)(2,8)(4,13)(5,14,20,30)(6,15)(7,16)(9,22)(10,12,17,27)(11,23)(18,29)(19,21,24,26)(25,28,31,32);(1,4,6,18,7,2,9,11)(3,13,15,29,16,8,22,23)(5,17,19,31,20,10,24,25)(12,26,28,14,27,21,32,30);(2,4)(8,13)(10,17)(11,18)(12,27)(23,29)(25,31)(28,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=i,b^a=be,d^a=di,e^a=ei,g^a=ghi,e^b=eh,h^b=hi,d^c=di,g^e=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72236884, #72237550, #72237552, … |
| maximal normal subgroup of | #72236884, #72237550, #72237552, … |
| maximal quotient of | #72312728, C2×#493042, #72337511 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400238);