Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
6406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(9,17)(13,18)(14,20)(16,19)(21,22)(23,28)(24,25)(26,34)(27,30)(29,35)(31,33)(32,36);(1,2)(3,4)(5,6)(7,8)(9,13)(10,11)(12,15)(14,16)(17,18)(19,20)(21,23,27,24)(22,28,30,25)(26,31,35,32)(29,36,34,33);(6,11)(8,12)(13,18)(16,19)(21,24,27,23)(22,28,30,25)(26,32,35,31)(29,36,34,33);(5,7)(6,12)(8,11)(9,14)(10,15)(13,19)(16,18)(17,20)(21,25,27,28)(22,24,30,23)(26,33,35,36)(29,32,34,31);(5,8,10,12)(6,7,11,15)(9,16,17,19)(13,14,18,20)(21,26)(22,29)(23,31)(24,32)(25,33)(27,35)(28,36)(30,34);(5,9,10,17)(6,13,11,18)(7,14,15,20)(8,16,12,19)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622864);