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 | |
6932 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)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(24,32)(27,34)(30,36)(33,35);(1,2)(3,4)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(6,10)(11,13)(12,17)(18,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7616940);