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 | |
7004 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)(8,16)(11,18)(14,20)(17,19)(21,22,26,30)(23,29,25,27)(24,28,34,32)(31,36,33,35);(1,2)(3,4)(5,6,9,12)(7,10,15,13)(8,11,16,18)(14,17,20,19);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,27)(24,32)(25,29)(26,30)(28,34)(31,36)(33,35);(21,23,26,25)(22,27,30,29)(24,31,34,33)(28,35,32,36);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,24)(22,28)(23,31)(25,33)(26,34)(27,35)(29,36)(30,32);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,25,26,23)(22,29,30,27)(24,33,34,31)(28,36,32,35)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,f^2=h,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, 7549835);