Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14,18,15)(16,20,17,19)(21,22)(23,29)(24,26)(25,32)(27,30)(28,33)(31,35)(34,36);(7,12)(9,11)(13,15,18,14)(16,19,17,20)(22,24)(23,30)(25,33)(35,36);(5,6)(7,9)(8,10)(11,12)(13,16,18,17)(14,19,15,20)(21,23)(22,27)(24,29)(25,34)(26,30)(28,36)(31,33)(32,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(21,24,26,22)(23,29,30,27)(25,28,33,32)(31,35,34,36);(5,7,8,12)(6,9,10,11)(13,17,18,16)(14,20,15,19)(21,25,26,33)(22,28,24,32)(23,31,30,34)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3946092);