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 | |
5368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,14)(11,12,15,16)(17,18,22,23)(19,27,30,24)(20,25,31,29)(21,28,32,26);(1,2)(3,4)(5,7)(6,8)(10,14)(11,15)(18,23)(20,31)(24,27)(26,28);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,22,30)(18,24,23,27)(20,26,31,28)(21,25,32,29);(3,8)(4,6)(9,12)(10,11)(13,16)(14,15)(17,19)(18,24)(20,26)(21,29)(22,30)(23,27)(25,32)(28,31);(1,4)(2,3)(5,6)(7,8)(17,20)(18,25)(19,28)(21,24)(22,31)(23,29)(26,30)(27,32);(1,2)(3,4)(5,7)(6,8)(17,21)(18,26)(19,29)(20,24)(22,32)(23,28)(25,30)(27,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,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, 8875733);