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 | |
5244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(10,12)(14,16)(17,18)(19,25)(20,23)(21,29)(22,24)(26,30)(27,31)(28,32);(1,3)(2,5)(4,8)(6,7)(11,15)(14,16)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32);(1,2,6,4)(3,8,7,5)(9,10,13,12)(11,14,15,16)(18,24)(21,30)(23,25)(27,28);(2,4)(3,7)(17,18)(19,25)(20,23)(21,26)(22,24)(27,32)(28,31)(29,30);(1,4,6,2)(3,8,7,5)(9,11)(10,14)(12,16)(13,15)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31);(1,5,6,8)(2,7,4,3)(9,12,13,10)(11,16,15,14)(17,21,22,30)(18,26,24,29)(19,27,20,28)(23,31,25,32)Polycyclic presentation:
2^9|b^2=g,c^2=gh,e^2=gi,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, 8841958);