Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(18,21)(19,23)(25,26)(27,31)(28,30)(29,32);(2,5)(6,8)(9,11,14,16)(10,15,12,13)(19,23)(22,24)(25,26)(27,31)(28,30)(29,32);(2,5)(3,7)(10,12)(13,15)(17,18,20,21)(19,22,23,24)(25,26,28,30)(27,29,32,31);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,13,16,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,8)(4,5)(6,7)(9,13)(10,16)(11,12)(14,15)(17,19)(18,22)(20,23)(21,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=ghi,b^2=fghi,c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72326581, #72326598, #72326638, … |
| maximal normal subgroup of | #72326581, #72326598, #72326638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4098063);