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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4681 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)(11,15)(12,13)(14,16)(17,18,22,27)(19,25,28,23)(20,24,26,21)(29,30,31,32);(1,2,5,4)(3,7,8,6)(10,13)(14,15)(19,28)(23,25)(29,31)(30,32);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,23,27,25)(20,31,26,29)(21,32,24,30);(3,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,20,22,26)(18,24,27,21)(19,29,28,31)(23,32,25,30);(1,3)(2,7)(4,6)(5,8)(17,19)(18,25)(20,31)(21,30)(22,28)(23,27)(24,32)(26,29);(1,4,5,2)(3,6,8,7)(17,21,22,24)(18,26,27,20)(19,30,28,32)(23,31,25,29)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gi,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8935362);