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 | #37782 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,5,4,7)(10,13)(14,16)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31);(1,3,6,4)(2,7,8,5)(9,10,12,13)(11,14,15,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32);(2,8)(3,4)(18,20)(24,26)(25,28)(31,32);(1,2,6,8)(3,7,4,5)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31);(1,4,6,3)(2,5,8,7)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32);(1,5)(2,3)(4,8)(6,7)(9,11,12,15)(10,14,13,16)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7617107);