Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2126 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)(11,12,16,15)(17,18)(19,27)(20,30)(21,29)(22,23)(24,28)(25,31)(26,32);(6,8)(9,11,14,16)(10,15,13,12)(18,23)(19,28)(20,31)(21,32);(5,6,7,8)(17,19)(18,24)(20,29)(21,25)(22,28)(23,27)(26,31)(30,32);(1,2,3,4)(9,12)(10,11)(13,16)(14,15)(17,20)(18,25)(19,29)(21,24)(22,31)(23,30)(26,28)(27,32);(9,13,14,10)(11,15,16,12)(17,21)(18,26)(19,25)(20,27)(22,32)(23,29)(24,31)(28,30)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=gh,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2830356);