Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1722 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)(3,5)(4,7)(6,8)(10,12)(11,15)(17,18,23,22)(19,20,24,21)(25,26)(27,28)(29,31)(30,32);(2,3)(6,7)(9,10,13,12)(11,16,15,14)(17,19,23,24)(18,21,22,20)(25,27,30,29)(26,31,32,28);(4,8)(6,7)(10,12)(14,16)(17,20,23,21)(18,19,22,24)(25,28,30,31)(26,27,32,29);(1,3,5,2)(4,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,21,23,20)(18,24,22,19)(25,26)(27,28)(29,31)(30,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,14,15,16)(17,22,23,18)(19,21,24,20)(25,29,30,27)(26,28,32,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fgi,d^2=fgh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038402);