Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(7,9)(8,11)(21,22)(23,30)(24,28)(25,33)(26,27)(29,35)(31,32)(34,36);(5,6,10,12)(7,11,9,8)(14,17)(18,20)(22,27)(23,28)(29,33)(31,36);(5,7,10,9)(6,11,12,8)(13,14,16,17)(15,18,19,20)(21,23)(22,24)(25,31)(26,28)(27,30)(29,32)(33,34)(35,36);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(21,24)(22,28)(23,27)(25,34)(26,30)(29,31)(32,35)(33,36);(5,9,10,7)(6,8,12,11)(13,15,16,19)(14,18,17,20)(21,25,26,35)(22,29,27,33)(23,31,28,36)(24,32,30,34)Polycyclic presentation:
2^9|b^2=fg,c^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1305623);