Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,13)(14,15)(17,18,21,24)(19,20,23,22);(3,8)(5,7)(9,10,14,15)(11,16,12,13)(19,23)(20,22)(25,26,29,32)(27,30,28,31);(1,3)(2,5)(4,8)(6,7)(9,11,14,12)(10,13,15,16)(17,19,21,23)(18,22,24,20)(25,26)(27,30)(28,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,12,14,11)(10,13,15,16)(17,20,21,22)(18,23,24,19)(25,27,29,28)(26,30,32,31);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=gi,d^2=hi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2317948);