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 |
| prime radical | C2×#2699 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2761 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)(9,10,13,11)(12,14,16,15)(17,18)(19,27)(20,30)(21,29)(22,23)(24,28)(25,31)(26,32);(5,6,7,8)(10,11)(14,15)(18,23)(19,28)(25,30)(26,29);(5,6)(7,8)(9,10,13,11)(12,15,16,14)(17,19)(18,24)(20,29)(21,25)(22,28)(23,27)(26,31)(30,32);(1,2,3,4)(9,11,13,10)(12,14,16,15)(17,20,22,31)(18,25,23,30)(19,29,28,26)(21,27,32,24);(9,12,13,16)(10,14,11,15)(17,21)(18,26)(19,30)(20,24)(22,32)(23,29)(25,28)(27,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fg,d^2=fgi,e^2=fg,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, 3372568);