Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1895 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,13,11)(12,14,16,15)(17,18)(19,22)(20,24)(21,23)(25,26,28,30)(27,31,32,29);(2,5)(6,8)(10,11)(14,15)(17,19,21,24)(18,22,23,20)(25,26,28,30)(27,29,32,31);(3,7)(6,8)(9,10,13,11)(12,15,16,14)(17,19)(18,20)(21,24)(22,23)(25,27)(26,29)(28,32)(30,31);(1,2)(3,8)(4,5)(6,7)(9,11,13,10)(12,14,16,15)(17,19)(18,20)(21,24)(22,23)(25,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,20)(18,19)(21,22)(23,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=fgi,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3454323);