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 | |
6982 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)(7,12)(8,10)(9,16)(11,14)(13,17)(15,18)(19,20)(21,22,26,29)(23,24,27,31)(25,33,34,28)(30,36,35,32);(1,2,3,4)(5,7)(6,11)(8,12)(9,15)(10,14)(13,18)(16,19)(17,20)(23,27)(24,31)(30,35)(32,36);(6,8)(7,14)(13,16)(15,20)(21,23)(22,24)(25,30)(26,27)(28,32)(29,31)(33,36)(34,35);(5,7)(6,12)(8,11)(9,15)(10,14)(13,19)(16,18)(17,20);(5,8,10,6)(7,11,14,12)(9,16,17,13)(15,18,20,19)(21,24)(22,27)(23,29)(25,32)(26,31)(28,35)(30,33)(34,36);(5,9,10,17)(6,13,8,16)(7,15,14,20)(11,18,12,19)(21,25,26,34)(22,28,29,33)(23,30,27,35)(24,32,31,36)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,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7627696);