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 | |
6640 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)(7,11)(8,10)(9,16)(12,17)(13,14)(15,20)(18,19)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(1,2,3,4)(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,33)(35,36);(6,8)(7,13)(12,16)(15,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622731);