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 | |
6314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6)(7,12)(8,16)(9,14)(10,11)(13,18)(15,17)(19,20)(21,22)(23,27)(24,25)(26,30)(28,33)(29,34)(31,35)(32,36);(1,2,3,4)(21,23)(22,27)(24,31)(25,35)(26,33)(28,30)(29,32)(34,36);(6,11)(7,14)(13,16)(15,20)(21,23,26,33)(22,28,30,27)(24,31,34,36)(25,32,29,35);(5,7,10,14)(6,12,11,9)(8,15,18,20)(13,19,16,17)(21,24)(22,29)(23,31)(25,30)(26,34)(27,32)(28,35)(33,36);(5,8,10,18)(6,13,11,16)(7,15,14,20)(9,17,12,19)(21,23)(22,28)(24,31)(25,32)(26,33)(27,30)(29,35)(34,36);(5,9,10,12)(6,14,11,7)(8,17,18,19)(13,20,16,15)(21,25,26,29)(22,24,30,34)(23,32,33,35)(27,36,28,31)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650658);