Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| 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 | |
2125 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)(6,9)(7,11)(15,19)(18,20)(22,24)(23,29)(25,33)(34,36);(1,2)(3,4)(6,9)(10,12)(14,17)(18,20)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34);(13,14,16,17)(15,18,19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,6)(7,10)(8,9)(11,12)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,18,17,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6039715);