Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021363 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,23,32,24,27,22,26)(19,28,29,20,30,21,25,31);(3,9)(4,12)(6,13)(7,11)(17,19)(18,20)(21,26)(22,29)(23,25)(24,30)(27,31)(28,32);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,23,21,24,31,22,28)(18,25,26,30,27,29,32,19);(3,9)(6,13)(10,15)(14,16)(17,21,22,20,24,28,23,31)(18,25,26,30,27,29,32,19);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,22,24,23)(18,26,27,32)(19,29,30,25)(20,21,31,28)Polycyclic presentation:
2^9|a^2=fhi,c^2=gh,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104034);