Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,18,20)(15,19,16,17)(22,26)(23,29)(24,32)(34,36);(2,3)(13,15,18,16)(14,17,20,19)(22,26)(27,30)(28,33)(34,36);(5,7,8,12)(6,9,10,11)(13,16,18,15)(14,19,20,17)(21,22,25,26)(23,30,29,27)(24,28,32,33)(31,36,35,34);(1,3,4,2)(5,6)(7,11)(8,10)(9,12)(13,17,18,19)(14,16,20,15)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)Polycyclic presentation:
2^9|a^2=gh,b^2=fgi,c^2=g,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4837471);