Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1874 conjugacy classes of subgroups; only the 510 normal subgroups are shown.
This group has 1874 conjugacy classes of subgroups — too many to draw. Only its 510 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(10,13)(11,15)(18,20)(19,23)(25,26,28,31)(27,30,32,29);(11,15)(14,16)(18,20)(22,24)(25,27,28,32)(26,29,31,30);(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,23,24)(25,26,28,31)(27,30,32,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,22,20,24)(25,27)(26,30)(28,32)(29,31);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,24,23,22)(25,26)(27,29)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038423);