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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3532 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)(9,10)(11,14)(12,16)(13,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31);(5,6,7,8)(9,11,13,16)(10,14,15,12)(18,22)(19,26)(24,28)(27,29);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,23)(20,28)(21,22)(24,30)(25,26)(27,32)(29,31);(1,2)(3,4)(9,11)(10,12)(13,16)(14,15)(17,19,21,26)(18,23,22,25)(20,29,30,27)(24,32,28,31);(9,12)(10,11)(13,14)(15,16)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)Polycyclic presentation:
2^9|b^2=fh,d^2=f,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, 3241971);