Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of #94823 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7529 |
| commutator | C82 | C2×C4 |
| Frattini, cosocle | #1712 | C22 |
| prime radical | #12515 | C2 |
| LCS | G ▷ C82 ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1712 ◁ G | |
| LECS | G ▷ #1712 ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,13,14,8)(4,10,19,9)(5,11,6,22)(7,12)(15,29,31,28)(16,23,18,24)(17,30,32,25)(20,27,21,26);(1,3)(2,8)(4,18)(5,16)(6,15)(7,17)(9,23)(10,13)(11,28)(12,25)(14,19)(20,32)(21,31)(22,24)(26,30)(27,29)Polycyclic presentation:
2^9|a^2=d,c^2=fg,e^2=gh,f^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,h^a=hi,c^b=cfgh,d^b=def,e^b=eghi,f^b=fhi,g^b=gi,d^c=dfg,e^d=eghi,f^d=fhi,g^d=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2019);