#244032

order	512
nilpotency class	3
isoclinism	isoclinic class of #233431
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

746 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,c^2=h,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151228);