Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,9)(7,11)(14,16)(18,20)(21,22,26,30)(23,31,34,27)(24,33,35,28)(25,29,36,32);(2,4)(6,9)(10,12)(15,19)(18,20)(21,23,26,34)(22,27,30,31)(24,32,35,29)(25,33,36,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,18,19,20)(21,23)(22,27)(24,29)(25,33)(26,34)(28,36)(30,31)(32,35);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,24,26,35)(22,28,30,33)(23,32,34,29)(25,31,36,27);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,20,19,18)(21,25)(22,29)(23,33)(24,27)(26,36)(28,34)(30,32)(31,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2681144);