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 | |
6755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6)(7,12)(8,16)(9,13)(10,15)(11,17)(14,18)(19,20);(1,2)(3,4)(22,26)(27,33)(28,34)(35,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,22)(23,27)(24,28)(25,26)(29,35)(30,33)(31,36)(32,34);(5,8,9,17)(6,11,13,16)(7,14,15,20)(10,18,12,19);(5,7)(6,12)(8,14)(9,15)(10,13)(11,19)(16,18)(17,20)(21,23,25,30)(22,27,26,33)(24,31,32,29)(28,36,34,35);(5,6)(7,10)(8,11)(9,13)(12,15)(14,18)(16,17)(19,20)(21,24,25,32)(22,28,26,34)(23,29,30,31)(27,35,33,36)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7560648);