#5065289

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	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

1980 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,5,6)(3,4,8,7)(9,10,14,16)(11,13,12,15)(18,20)(19,23)(25,26)(27,30)(28,31)(29,34)(32,37)(33,35)(36,39)(38,40);(2,6)(3,8)(9,11)(10,15)(12,14)(13,16)(17,18,21,20)(19,24,23,22)(26,27)(28,35)(29,37)(38,39);(3,8)(4,7)(11,12)(13,15)(18,20)(19,23)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,3,5,8)(2,7,6,4)(9,12,14,11)(10,15,16,13)(17,19,21,23)(18,22,20,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,4,5,7)(2,8,6,3)(9,13,14,15)(10,11,16,12)(17,20,21,18)(19,24,23,22)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fhi,c^2=g,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4972485);