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 | |
5029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,13,14)(11,12,15,16)(17,18)(19,25)(20,23)(21,26)(22,24)(27,32)(28,31)(29,30);(1,3,5,8)(2,6,4,7)(17,19,23,26)(18,21,20,25)(22,30,31,27)(24,32,28,29);(10,14)(11,15)(18,20)(19,26)(22,31)(29,32);(1,2,5,4)(3,6,8,7)(9,11)(10,12)(13,15)(14,16)(17,20,23,18)(19,21,26,25)(22,28,31,24)(27,29,30,32);(9,12)(10,11)(13,16)(14,15)(17,21)(18,19)(20,26)(22,29)(23,25)(24,27)(28,30)(31,32);(1,4,5,2)(3,7,8,6)(17,22,23,31)(18,24,20,28)(19,27,26,30)(21,29,25,32)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713872);