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 |
| prime radical | #34641 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5028 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(18,22)(19,25)(24,29)(26,31);(1,3,5,4)(2,6,7,8)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,3,5,4)(2,6,7,8)(9,11,13,14)(10,12,15,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,4,5,3)(2,6,7,8)(9,10)(11,12)(13,15)(14,16);(1,2)(3,6)(4,8)(5,7)(9,12)(10,14)(11,15)(13,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)Polycyclic presentation:
2^9|b^2=g,c^2=hi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7563956);