Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#106 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #103 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(18,23,24,25)(19,27)(20,29)(21,28)(26,32,30,31);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,18,20,25,21,26,29,32,22,24,19,23,28,30,27,31);(3,11)(7,15)(10,16)(12,13)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,c^b=ch,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72319517, #72319538, #72319544, … |
| maximal normal subgroup of | #72319517, #72319538, #72319544, … |
| maximal quotient of | C2×#126836 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34032);