Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2941 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,15)(14,16)(17,18,21,22)(19,20,23,24)(26,28)(30,31);(1,3,4,8)(2,5,6,7)(10,11)(12,16)(18,22)(19,23)(25,26)(27,31)(28,29)(30,32);(2,6)(5,7)(9,10,13,11)(12,15,16,14)(17,19)(18,20)(21,23)(22,24)(25,27,29,32)(26,30,28,31);(1,3)(2,5)(4,8)(6,7)(9,11,13,10)(12,14,16,15)(17,19)(18,20)(21,23)(22,24)(25,28,29,26)(27,31,32,30);(1,2)(3,7)(4,6)(5,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,23)(19,22)(21,24)(25,27)(26,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=fhi,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4024787);