Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2744 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)(3,8)(4,5)(6,7)(11,15)(14,16)(18,20)(19,23)(26,28)(30,32);(1,3,6,4)(2,5,7,8)(9,10,13,12)(11,14,15,16)(17,18,21,20)(19,24,23,22)(27,31)(30,32);(2,7)(3,4)(19,23)(22,24)(25,26,29,28)(27,30,31,32);(1,4,6,3)(2,5,7,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,23)(18,22,20,24)(25,27)(26,30)(28,32)(29,31);(1,5)(2,4)(3,7)(6,8)(9,12,13,10)(11,16,15,14)(17,20,21,18)(19,22,23,24)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2708273);