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 | |
6926 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)(7,8)(9,17)(10,11)(12,15)(13,18)(14,19)(16,20)(21,22,26,27)(23,24,28,32)(25,33,34,29)(30,36,35,31);(1,2)(3,4)(22,27)(23,28)(29,33)(30,35);(6,11)(8,12)(13,17)(16,19)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,36)(34,35);(5,7)(6,12)(8,11)(9,14)(10,15)(13,19)(16,17)(18,20);(5,8,10,12)(6,7,11,15)(9,16,18,19)(13,14,17,20)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,33)(34,36);(5,9,10,18)(6,13,11,17)(7,14,15,20)(8,16,12,19)(21,25,26,34)(22,29,27,33)(23,30,28,35)(24,31,32,36)Polycyclic presentation:
2^9|a^2=i,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550641);