Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
2114 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)(5,6,9,7)(8,11,12,10)(15,19)(18,20)(21,22)(23,30)(24,28)(25,29)(26,27)(31,36)(32,33)(34,35);(5,6,9,7)(8,11,12,10)(14,17)(18,20)(22,27)(23,28)(29,35)(31,33);(6,7)(10,11)(13,14,16,17)(15,18,19,20)(21,23)(22,24)(25,33)(26,28)(27,30)(29,36)(31,34)(32,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(21,24)(22,28)(23,27)(25,32)(26,30)(29,33)(31,35)(34,36);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,34)(22,29,27,35)(23,31,28,33)(24,32,30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=gh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5382312);