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 | |
4892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,7)(9,10,12,14)(11,13,15,16)(17,18,23,25)(19,24,20,22)(21,29,30,26)(27,32,28,31);(1,2,5,4)(3,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19,23,20)(18,24,25,22)(21,27,30,28)(26,31,29,32);(1,2,5,4)(3,7,8,6)(18,25)(19,20)(21,30)(31,32);(1,3,5,8)(2,6,4,7)(9,11,12,15)(10,13,14,16)(17,20,23,19)(18,24,25,22)(21,28,30,27)(26,31,29,32);(1,4,5,2)(3,7,8,6)(17,21)(18,26)(19,27)(20,28)(22,32)(23,30)(24,31)(25,29);(17,22,23,24)(18,19,25,20)(21,31,30,32)(26,28,29,27)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=hi,d^2=gi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7896827);