#413852

order	512
nilpotency class	3
isoclinism	isoclinic class of #411585
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=g,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^c=fi

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, 321048);