Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #59693 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18)(19,27)(20,22)(21,24)(23,25)(26,31)(28,29)(30,32);(1,2,5,4)(3,6,7,8)(10,11)(14,15)(17,19,23,21)(18,24,25,27)(20,30,31,28)(22,29,26,32);(2,4)(6,8)(9,11,13,10)(12,15,16,14)(18,25)(22,26)(24,27)(29,32);(3,7)(6,8)(9,12,13,16)(10,14,11,15)(17,20)(18,26)(19,28)(21,30)(22,25)(23,31)(24,29)(27,32);(1,3)(2,6)(4,8)(5,7)(17,21,23,19)(18,27,25,24)(20,28,31,30)(22,32,26,29);(1,4,5,2)(3,8,7,6)(17,22,23,26)(18,20,25,31)(19,29,21,32)(24,30,27,28)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8232712);