Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| 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 | |
1652 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,12)(15,16)(17,18,21,25)(19,22,27,31)(20,23,28,24)(26,29,32,30);(1,3)(2,7)(4,13)(5,15)(6,11)(8,12)(9,16)(10,14)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(3,11)(7,16)(10,12)(13,15)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,2)(3,12)(4,8)(5,14)(6,9)(7,15)(10,11)(13,16);(1,4,6,5)(2,8,9,14)(3,13,11,15)(7,12,16,10)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=df,f^b=fh,g^c=gi,f^d=fh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265491);