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 | |
6572 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)(5,6,10,14)(7,13,9,11)(8,12,18,16)(15,20,17,19)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33);(1,2)(3,4)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(5,7,10,9)(6,11,14,13)(8,15,18,17)(12,19,16,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,6)(7,11)(8,16)(9,13)(10,14)(12,18)(15,20)(17,19)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,20)(14,16);(5,9,10,7)(6,13,14,11)(8,17,18,15)(12,20,16,19)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7553112);