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 | |
6940 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)(5,6)(7,14)(8,16)(9,11)(10,13)(12,18)(15,19)(17,20);(1,2,3,4)(21,22)(23,28)(24,27)(25,29)(26,31)(30,34)(32,35)(33,36);(5,7,10,9)(6,11,13,14)(8,15,18,17)(12,19,16,20)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,16)(14,20)(21,24,25,32)(22,27,29,35)(23,30,31,36)(26,33,28,34);(5,9,10,7)(6,11,13,14)(8,17,18,15)(12,19,16,20)(21,23)(22,28)(24,30)(25,31)(26,29)(27,34)(32,36)(33,35);(5,6)(7,11)(8,12)(9,14)(10,13)(15,19)(16,18)(17,20)(21,22)(23,26)(24,27)(25,29)(28,31)(30,33)(32,35)(34,36)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=i,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, 7562615);