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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10,12,13)(11,15,16,14)(17,18,21,24)(19,20,23,22)(25,26,28,30)(27,31,32,29);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,29)(28,30)(31,32);(2,7)(6,8)(10,13)(14,15)(19,23)(20,22)(25,26)(27,31)(28,30)(29,32);(1,4,5,3)(2,6,7,8)(9,11,12,16)(10,14,13,15)(17,18)(19,20)(21,24)(22,23)(25,26)(27,31)(28,30)(29,32);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,13)(14,16)(17,20)(18,23)(19,24)(21,22)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=ghi,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, 3408492);