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 | |
7276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6)(7,14)(8,16)(9,11)(10,13)(12,18)(15,19)(17,20)(21,22,26,29)(23,27,30,34)(24,25,28,33)(31,32,35,36);(1,2)(3,4)(23,30)(27,34)(31,35)(32,36);(5,7,10,9)(6,11,13,14)(8,15,18,17)(12,19,16,20)(21,23)(22,27)(24,31)(25,32)(26,30)(28,35)(29,34)(33,36);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,16)(14,20);(5,9,10,7)(6,11,13,14)(8,17,18,15)(12,19,16,20)(21,24)(22,25)(23,31)(26,28)(27,32)(29,33)(30,35)(34,36);(5,6)(7,11)(8,12)(9,14)(10,13)(15,19)(16,18)(17,20)(21,25)(22,28)(23,32)(24,29)(26,33)(27,35)(30,36)(31,34)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561185);