Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994171 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32);(1,3,4,8)(2,5,6,7)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(10,13)(11,16)(14,15)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,31,32,30);(3,8)(5,7)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31);(18,24)(22,25)(23,32)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7032911);