Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C22 | C2×D42 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,9)(3,10,13,11)(4,8,5,14)(7,15,16,12)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32);(1,3,6,13)(2,7,9,16)(4,12,5,15)(8,11,14,10);(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(1,2)(3,11)(4,8)(5,14)(6,9)(7,15)(10,13)(12,16);(1,4,6,5)(2,8,9,14)(3,12,13,15)(7,11,16,10)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,d^b=df,f^b=fh,g^c=gi,f^d=fh98 conjugacy classes — view the full character table.
| maximal subgroup of | #72283267, #72283381, #72283393, … |
| maximal normal subgroup of | #72283267, #72283381, #72283393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265618);