Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105838 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,24,22,26,20,25)(19,27,30,32,28,23,29,31);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23,26,27)(20,29)(21,30)(22,28)(24,31,25,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,24)(20,21)(23,31)(25,26)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,f^b=fh,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72263987, #72263989, #72263998, … |
| maximal normal subgroup of | #72263987, #72263989, #72263998, … |
| maximal quotient of | #72264364 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13046);