Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,13,14)(11,12,16,15)(17,18,21,20)(19,22,24,23)(25,26,29,30)(27,32,28,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,24)(18,22,20,23)(26,30)(31,32);(3,8)(6,7)(10,14)(11,16)(18,20)(22,23)(25,27,29,28)(26,31,30,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20,21,18)(19,23,24,22)(25,28,29,27)(26,31,30,32);(1,4,5,2)(3,7,8,6)(9,12,13,15)(10,11,14,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,30)Polycyclic presentation:
2^9|a^2=ghi,b^2=fgh,c^2=fhi,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5788105);