Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 8388608 |
| 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 | |
2010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(6,9)(7,11)(14,18)(15,20)(21,22,25,28)(23,29,30,26)(24,32,33,27)(31,34,36,35);(1,2,4,3)(13,14)(15,19)(16,20)(17,18)(23,30)(26,29)(31,36)(34,35);(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,15,17,20)(14,19,18,16)(21,23,25,30)(22,26,28,29)(24,31,33,36)(27,34,32,35);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,16,17,19)(14,20,18,15)(21,22)(23,26)(24,27)(25,28)(29,30)(31,34)(32,33)(35,36);(21,24,25,33)(22,27,28,32)(23,31,30,36)(26,34,29,35)Polycyclic presentation:
2^9|a^2=h,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=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215572);