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 | |
4921 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,29)(19,27,21,24)(20,26,31,25)(22,30,32,28);(1,3,5,6)(2,4,7,8)(9,10,12,14)(11,13,16,15)(17,19,23,21)(18,24,29,27)(20,30,31,28)(22,25,32,26);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,23,31)(18,25,29,26)(19,30,21,28)(22,24,32,27);(3,6)(4,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,28)(21,30)(22,27)(23,31)(24,32)(25,29);(1,4)(2,3)(5,8)(6,7)(17,21,23,19)(18,27,29,24)(20,28,31,30)(22,26,32,25);(1,2)(3,4)(5,7)(6,8)(17,22,23,32)(18,28,29,30)(19,25,21,26)(20,27,31,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840954);