Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(7,11)(10,12)(16,20)(18,19)(21,22,25,28)(23,26,29,32)(24,27,31,34)(30,33,35,36);(1,2,3,4)(6,9)(10,12)(13,14,17,15)(16,18,20,19)(23,29)(26,32)(30,35)(33,36);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(1,2)(3,4)(13,16)(14,18)(15,19)(17,20)(21,23,25,29)(22,26,28,32)(24,30,31,35)(27,33,34,36);(5,7)(6,10)(8,11)(9,12)(21,24,25,31)(22,27,28,34)(23,30,29,35)(26,33,32,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4363430);