Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2050 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)(6,10)(7,8)(13,14,16,18)(15,17,19,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,33)(32,36)(34,35);(5,6,9,10)(7,12,8,11)(15,19)(17,20)(22,27)(23,29)(30,35)(31,36);(5,7,9,8)(6,11,10,12)(13,15)(14,17)(16,19)(18,20)(21,23,26,29)(22,28,27,24)(25,31,34,36)(30,33,35,32);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,17)(16,18)(19,20)(21,24,26,28)(22,29,27,23)(25,33,34,32)(30,31,35,36);(5,8,9,7)(6,11,10,12)(21,25,26,34)(22,30,27,35)(23,31,29,36)(24,32,28,33)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2508830);