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 | |
6396 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)(7,13)(8,16)(9,10)(11,14)(12,17)(15,18)(19,20)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(1,2,3,4)(5,6,9,10)(7,11,14,13)(8,12,17,16)(15,18,20,19);(6,10)(7,14)(12,16)(15,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,7,9,14)(6,11,10,13)(8,15,17,20)(12,18,16,19)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35);(5,8,9,17)(6,12,10,16)(7,15,14,20)(11,18,13,19)Polycyclic presentation:
2^9|a^2=hi,b^2=g,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, 7622718);