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 | |
6579 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)(5,6,10,12)(7,11,13,8)(9,14,18,19)(15,17,20,16)(22,27)(23,28)(25,33)(31,36);(1,2)(3,4)(5,7,10,13)(6,11,12,8)(9,15,18,20)(14,17,19,16)(21,22,26,27)(23,24,28,32)(25,29,33,34)(30,31,35,36);(6,12)(7,13)(14,19)(15,20)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,35)(34,36);(5,6)(7,11)(8,13)(9,14)(10,12)(15,17)(16,20)(18,19)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,34)(33,36);(5,8,10,11)(6,13,12,7)(9,17,18,16)(14,15,19,20)(21,25)(22,29)(23,30)(24,31)(26,33)(27,34)(28,35)(32,36);(5,9,10,18)(6,14,12,19)(7,15,13,20)(8,16,11,17)Polycyclic presentation:
2^9|a^2=i,b^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7827339);