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 | |
6758 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)(15,16)(18,20)(22,27)(23,28)(25,33)(31,36);(1,2,3,4)(5,6)(7,14)(8,11)(9,12)(10,13)(15,20)(16,18)(17,19);(5,7,10,8)(6,11,13,14)(9,15,17,16)(12,18,19,20)(21,22,26,27)(23,24,28,32)(25,29,33,34)(30,31,35,36);(5,6)(7,11)(8,14)(9,12)(10,13)(15,18)(16,20)(17,19)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,35)(34,36);(5,8,10,7)(6,11,13,14)(9,16,17,15)(12,18,19,20)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,34)(33,36);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,25)(22,29)(23,30)(24,31)(26,33)(27,34)(28,35)(32,36)Polycyclic presentation:
2^9|b^2=g,c^2=hi,e^2=i,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, 7553541);