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 | |
6659 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6,10,12)(7,11,14,9)(8,16,17,13)(15,18,20,19)(21,22)(23,29)(24,26)(25,35)(27,30)(28,31)(32,36)(33,34);(1,2)(3,4)(5,7,10,14)(6,11,12,9)(8,15,17,20)(13,19,16,18)(21,23,28,25)(22,29,31,35)(24,34,27,32)(26,33,30,36);(6,12)(7,14)(8,17)(18,19)(21,24,28,27)(22,26,31,30)(23,32,25,34)(29,36,35,33);(5,8)(6,13)(7,15)(9,18)(10,17)(11,19)(12,16)(14,20)(21,25,28,23)(22,29,31,35)(24,34,27,32)(26,36,30,33);(5,6)(7,11)(8,13)(9,14)(10,12)(15,19)(16,17)(18,20)(21,26)(22,24)(23,33)(25,36)(27,31)(28,30)(29,34)(32,35);(5,9,10,11)(6,7,12,14)(8,18,17,19)(13,15,16,20)(21,27,28,24)(22,30,31,26)(23,34,25,32)(29,33,35,36)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=hi,d^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7894679);