Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469971 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(18,22)(19,25)(20,27)(23,30)(24,28)(29,31);(11,12)(14,15)(17,18,21,24)(19,23,25,31)(20,28,27,22)(26,30,32,29);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,19,21,25)(18,23,24,31)(20,26,27,32)(22,29,28,30);(9,12,13,11)(10,15,16,14);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(19,25)(23,31)(26,32)(29,30)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396230);