Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,24)(21,23)(25,26,29,30)(27,31,32,28);(2,3)(6,7)(9,10,13,12)(11,14,16,15)(17,19,21,24)(18,22,23,20)(26,30)(27,32);(4,8)(6,7)(10,12)(14,15)(17,20)(18,19)(21,22)(23,24)(25,27,29,32)(26,31,30,28);(1,3,5,2)(4,6,8,7)(9,11,13,16)(10,14,12,15)(17,19)(18,20)(21,24)(22,23)(25,26)(27,31)(28,32)(29,30);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,14,16,15)(17,20)(18,19)(21,22)(23,24)(25,28,29,31)(26,27,30,32)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=gi,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, 3454338);