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) = C27 |
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 | |
1960 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)(7,10)(8,9)(11,12)(14,17)(18,20)(21,22,26,27)(23,28,30,35)(24,33,29,25)(31,32,36,34);(6,9)(10,12)(13,14,16,17)(15,18,19,20)(22,27)(23,30)(24,29)(32,34);(1,2,3,4)(7,11)(10,12)(15,19)(18,20)(21,23)(22,28)(24,31)(25,34)(26,30)(27,35)(29,36)(32,33);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24,26,29)(22,25,27,33)(23,31,30,36)(28,34,35,32);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,25)(22,29)(23,32)(24,27)(26,33)(28,31)(30,34)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=h,e^2=fh,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, 4293590);