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) = C26×C4 |
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 | |
2018 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)(7,12)(8,10)(9,11)(13,14)(15,16)(17,20)(18,19)(21,22,25,26)(23,27,29,33)(24,31,32,28)(30,36,35,34);(1,2,3,4)(14,19)(16,20)(22,26)(23,29)(28,31)(30,35);(5,7,10,9)(6,11,8,12)(13,15,18,17)(14,20,19,16)(21,23)(22,27)(24,30)(25,29)(26,33)(28,34)(31,36)(32,35);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,16,18,20)(14,15,19,17)(21,22)(23,27)(24,28)(25,26)(29,33)(30,34)(31,32)(35,36);(5,9,10,7)(6,11,8,12)(13,17,18,15)(14,20,19,16)(21,24,25,32)(22,28,26,31)(23,30,29,35)(27,34,33,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,c^2=g,d^2=g,e^2=gh,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, 4837545);