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 | |
6534 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)(5,6,10,11)(7,12,14,9)(8,16,18,13)(15,17,20,19)(21,22,25,29)(23,27,30,26)(24,32,33,28)(31,34,36,35);(1,2,3,4)(23,30)(26,27)(31,36)(34,35);(6,11)(7,14)(13,16)(15,20)(21,23,25,30)(22,26,29,27)(24,31,33,36)(28,34,32,35);(5,7)(6,12)(8,15)(9,11)(10,14)(13,19)(16,17)(18,20)(21,23)(22,27)(24,31)(25,30)(26,29)(28,35)(32,34)(33,36);(5,8,10,18)(6,13,11,16)(7,15,14,20)(9,17,12,19)(21,22)(23,27)(24,28)(25,29)(26,30)(31,35)(32,33)(34,36);(5,9,10,12)(6,7,11,14)(8,17,18,19)(13,15,16,20)(21,24,25,33)(22,28,29,32)(23,31,30,36)(26,34,27,35)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,e^2=hi,f^2=h,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, 7622059);