Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,14,12)(11,16,13,15)(17,18)(19,21)(20,22)(23,24)(27,31)(29,32);(1,2)(3,7)(4,6)(5,8)(9,10,14,12)(11,15,13,16)(17,18,20,22)(19,23,24,21)(25,26,28,30)(27,29,31,32);(3,8)(5,7)(9,11,14,13)(10,15,12,16)(17,18)(19,23)(20,22)(21,24);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,15,13,16)(25,27)(26,29)(28,31)(30,32);(1,3,4,8)(2,5,6,7)(9,13,14,11)(10,15,12,16)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,c^2=g,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393458);