Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2116 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)(7,8)(11,12)(15,18)(16,20)(21,22,26,27)(23,28,30,25)(24,32,33,29)(31,34,36,35);(6,10)(11,12)(13,14,17,19)(15,16,18,20)(22,27)(23,30)(24,33)(34,35);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,15,17,18)(14,16,19,20)(21,23)(22,28)(24,31)(25,27)(26,30)(29,35)(32,34)(33,36);(5,7,9,8)(6,11,10,12)(13,16)(14,18)(15,19)(17,20)(21,24,26,33)(22,29,27,32)(23,31,30,36)(25,34,28,35);(5,8,9,7)(6,12,10,11)(21,25,26,28)(22,30,27,23)(24,34,33,35)(29,36,32,31)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=g,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4293543);