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) = 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 | |
1958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(7,11)(10,12)(13,14,16,18)(15,19,20,17)(21,22)(23,28)(24,29)(25,33)(26,27)(30,35)(31,32)(34,36);(2,4)(5,6,9,8)(7,10,11,12)(13,14,16,18)(15,17,20,19)(22,27)(23,30)(24,33)(32,36);(1,2)(3,4)(13,14)(15,17)(16,18)(19,20)(21,23,26,30)(22,28,27,35)(24,34,33,31)(25,32,29,36);(5,7)(6,10)(8,12)(9,11)(13,14)(15,17)(16,18)(19,20)(21,24,26,33)(22,29,27,25)(23,31,30,34)(28,32,35,36);(5,8,9,6)(7,12,11,10)(13,15)(14,17)(16,20)(18,19)(21,25,26,29)(22,24,27,33)(23,32,30,36)(28,34,35,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=f,e^2=fgi,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, 2696869);