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 | #51568 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,13,15)(11,12,16,14);(1,3,5,8)(2,6,4,7)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31);(1,2,5,4)(3,6,8,7)(9,11,13,16)(10,14,15,12)(18,22)(19,25)(24,27)(26,31);(1,2,5,4)(3,6,8,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(9,12)(10,11)(13,14)(15,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32);(1,4,5,2)(3,7,8,6)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7644147);