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 | |
6814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(9,17)(12,19)(15,18)(16,20)(22,27)(23,25)(29,32)(31,33);(1,2,3,4)(5,6)(7,14)(8,11)(9,12)(10,13)(15,20)(16,18)(17,19)(21,22)(23,28)(24,29)(25,30)(26,27)(31,35)(32,34)(33,36);(5,7,10,11)(6,8,13,14)(9,15,17,18)(12,16,19,20)(21,23,26,25)(22,28,27,30)(24,31,34,33)(29,35,32,36);(5,6)(7,8)(9,12)(10,13)(11,14)(15,16)(17,19)(18,20)(21,22)(23,28)(24,32)(25,30)(26,27)(29,34)(31,36)(33,35);(5,8)(6,11)(7,13)(9,16)(10,14)(12,18)(15,19)(17,20)(21,24)(22,29)(23,31)(25,33)(26,34)(27,32)(28,35)(30,36);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,25,26,23)(22,30,27,28)(24,33,34,31)(29,36,32,35)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7552800);