Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(10,13)(11,15)(17,18,22,26)(19,23,24,20)(21,25,30,32)(27,29,31,28);(6,8)(11,15)(14,16)(17,19,22,24)(18,20,26,23)(21,27,30,31)(25,28,32,29);(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,29)(28,32)(30,31);(1,2)(3,4)(9,11)(10,14)(12,15)(13,16)(17,20,22,23)(18,24,26,19)(21,29,30,28)(25,27,32,31);(9,10)(11,14)(12,13)(15,16)(17,21,22,30)(18,25,26,32)(19,27,24,31)(20,28,23,29)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2643746);