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 | |
4979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,4,7,6)(9,10,13,15)(11,12,16,14)(18,23)(19,26)(21,29)(28,31);(1,3)(2,6)(4,8)(5,7)(9,11,13,16)(10,14,15,12)(17,18,22,23)(19,20,26,24)(21,25,29,30)(27,28,32,31);(1,4)(2,7)(3,8)(5,6)(9,11)(10,12)(13,16)(14,15)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28);(1,2,5,8)(3,4,7,6)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31);(1,3)(2,4)(5,7)(6,8)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,31)(30,32);(1,4)(2,7)(3,8)(5,6)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9949781);