Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | C2×#2543 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2769 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)(9,10,13,16)(11,12,15,14)(17,18)(19,25)(20,28)(21,27)(22,23)(24,29)(26,31)(30,32);(6,8)(9,11,13,15)(10,14,16,12)(18,23)(20,29)(21,25)(26,30);(5,6)(7,8)(9,12,13,14)(10,15,16,11)(17,19)(18,21)(20,30)(22,27)(23,25)(24,32)(26,29)(28,31);(1,2,3,4)(9,12)(10,11)(13,14)(15,16)(17,20)(18,24)(19,26)(21,31)(22,29)(23,28)(25,32)(27,30);(9,11)(10,12)(13,15)(14,16)(17,21)(18,19)(20,31)(22,25)(23,27)(24,26)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fg,d^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1520880);