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 | |
6628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6,12,16)(7,10,9,13)(8,15,11,14)(17,19,18,20)(21,22)(23,28)(24,25)(26,34)(27,30)(29,35)(31,33)(32,36);(1,2,3,4)(5,7)(6,13)(8,18)(9,12)(10,16)(11,17)(14,20)(15,19)(21,23,27,24)(22,28,30,25)(26,31,35,32)(29,36,34,33);(5,8,12,11)(6,14,16,15)(7,17,9,18)(10,20,13,19)(21,24,27,23)(22,28,30,25)(26,32,35,31)(29,36,34,33);(5,9,12,7)(6,13,16,10)(8,17,11,18)(14,20,15,19)(21,25,27,28)(22,24,30,23)(26,33,35,36)(29,32,34,31);(5,10,12,13)(6,7,16,9)(8,19,11,20)(14,18,15,17)(21,26,27,35)(22,29,30,34)(23,31,24,32)(25,33,28,36);(5,11,12,8)(6,15,16,14)(7,17,9,18)(10,20,13,19)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=h,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, 7622759);