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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(9,10,14,13)(11,16,12,15)(17,18,22,25)(19,26,23,20)(21,29,31,24)(27,28,32,30);(6,8)(9,11)(10,15)(12,14)(13,16)(17,19,22,23)(18,20,25,26)(21,27,31,32)(24,28,29,30);(5,6,7,8)(11,12)(15,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,30)(31,32);(1,2)(3,4)(9,12,14,11)(10,15,13,16)(17,20)(18,23)(19,25)(21,30)(22,26)(24,27)(28,31)(29,32);(9,13,14,10)(11,16,12,15)(17,21,22,31)(18,24,25,29)(19,27,23,32)(20,28,26,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=fg,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, 2643812);