Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2217 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)(5,6)(7,9)(8,11)(10,12)(14,19)(15,16)(21,22,26,29)(23,30,27,24)(25,33,34,28)(31,32,35,36);(6,7)(10,11)(13,14,18,19)(15,17,16,20)(23,27)(24,30)(31,35)(32,36);(5,6,9,7)(8,10,12,11)(13,15,18,16)(14,17,19,20)(21,23,26,27)(22,24,29,30)(25,31,34,35)(28,32,33,36);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,16,18,15)(14,17,19,20)(21,24)(22,27)(23,29)(25,32)(26,30)(28,35)(31,33)(34,36);(5,8)(6,10)(7,11)(9,12)(13,17)(14,16)(15,19)(18,20)(21,25,26,34)(22,28,29,33)(23,31,27,35)(24,32,30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4769669);