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 |
| prime radical | #35292 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6596 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)(8,16)(11,18)(14,20)(17,19)(21,22,25,28)(23,26,30,34)(24,27,32,36)(29,33,31,35);(1,2)(3,4)(5,6)(7,12)(8,11)(9,13)(10,15)(14,19)(16,18)(17,20);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,35)(32,36);(5,8,9,16)(6,11,13,18)(7,14,15,20)(10,17,12,19);(5,7)(6,12)(8,14)(9,15)(10,13)(11,19)(16,20)(17,18)(21,23,25,30)(22,26,28,34)(24,31,32,29)(27,35,36,33);(5,6)(7,10)(8,11)(9,13)(12,15)(14,17)(16,18)(19,20)(21,24,25,32)(22,27,28,36)(23,29,30,31)(26,33,34,35)Polycyclic presentation:
2^9|a^2=gh,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7564623);