Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #127312 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×H(3,ℤ/4) |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C4×M16 | C23 |
| socle | C2 | #7972 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
360 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)(3,8)(4,16,18,7)(5,11)(6,12)(9,19,21,30)(10,17)(13,26)(14,22,27,20)(15,25)(23,29,31,32)(24,28);(2,7,10,21,12,16,11,9)(3,13)(8,22,24,23,26,20,25,31)(14,27)(15,28)(29,32);(1,3,5,15,6,13,17,28)(2,8,11,25,12,26,10,24)(4,14,19,29,18,27,30,32)(7,20,9,23,16,22,21,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=ghi,f^2=h,g^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=ci,e^b=eh,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286336, #72287445, #72287450, … |
| maximal normal subgroup of | #72286336, #72287445, #72287450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34512);