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 | C2×#3511 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6760 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)(8,16)(12,19)(14,20)(17,18)(21,22,27,31)(23,25,29,36)(24,30,26,28)(32,35,33,34);(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,15)(14,18)(16,19)(17,20)(21,23,27,29)(22,25,31,36)(24,32,26,33)(28,34,30,35);(5,7)(6,10)(8,14)(9,15)(11,13)(12,17)(16,20)(18,19)(21,24,27,26)(22,28,31,30)(23,32,29,33)(25,34,36,35);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,22)(23,25)(24,28)(26,30)(27,31)(29,36)(32,34)(33,35);(5,6)(7,10)(8,12)(9,13)(11,15)(14,17)(16,19)(18,20)(21,25)(22,29)(23,31)(24,34)(26,35)(27,36)(28,33)(30,32);(5,8,9,16)(6,12,13,19)(7,14,15,20)(10,17,11,18)(21,26,27,24)(22,30,31,28)(23,33,29,32)(25,35,36,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,f^2=hi,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, 7554826);