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 | |
5309 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)(17,18,23,27)(19,26,21,24)(20,25,31,22)(28,29,30,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,13)(14,15)(17,19,23,21)(18,24,27,26)(20,28,31,30)(22,29,25,32);(3,8)(6,7)(10,15)(11,12)(17,20,23,31)(18,22,27,25)(19,28,21,30)(24,29,26,32);(1,3)(2,7)(4,6)(5,8)(9,11,14,12)(10,16,15,13)(17,20)(18,25)(19,28)(21,30)(22,27)(23,31)(24,32)(26,29);(1,3)(2,7)(4,6)(5,8)(9,12,14,11)(10,13,15,16)(17,21,23,19)(18,26,27,24)(20,30,31,28)(22,32,25,29);(1,4,5,2)(3,6,8,7)(9,13)(10,11)(12,15)(14,16)(17,22,23,25)(18,20,27,31)(19,29,21,32)(24,28,26,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=ghi,e^2=gi,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, 7644746);