Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368558 |
| 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 | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,11)(3,12,14,4)(5,10,6,9)(8,15,16,13)(17,18)(19,24)(20,25)(21,23)(22,27)(26,29)(28,31)(30,32);(1,3)(2,8)(4,10)(5,13)(6,15)(7,14)(9,12)(11,16)(17,19,20,27)(18,22,25,24)(21,28,29,32)(23,30,26,31);(1,3)(2,4)(5,15)(6,13)(7,14)(8,10)(9,16)(11,12)(18,23)(21,29)(22,30)(24,31)(25,26)(28,32);(1,4)(2,3)(5,8)(6,16)(7,12)(9,13)(10,15)(11,14)(17,19)(18,24)(20,27)(21,28)(22,25)(23,31)(26,30)(29,32);(1,5,7,6)(2,9,11,10)(3,13,14,15)(4,8,12,16)(19,27)(22,24)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275783);