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 | |
7000 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)(5,6)(7,14)(8,16)(9,11)(10,13)(12,18)(15,19)(17,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,35)(32,33)(34,36);(1,2)(3,4)(21,23,26,28)(22,25,29,30)(24,32,34,31)(27,35,36,33);(5,7,10,9)(6,11,13,14)(8,15,18,17)(12,19,16,20)(21,24,26,34)(22,27,29,36)(23,31,28,32)(25,33,30,35);(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)(21,22)(23,25)(24,27)(26,29)(28,30)(31,33)(32,35)(34,36);(5,6)(7,11)(8,12)(9,14)(10,13)(15,19)(16,18)(17,20)(21,25)(22,28)(23,29)(24,33)(26,30)(27,32)(31,36)(34,35)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=hi,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, 7562498);