#170404

order	512
nilpotency class	3
isoclinism	isoclinic class of #170381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

867 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77600);