#293200

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

748 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)(3,7)(4,6)(5,8)(9,10,15,18)(11,16,21,24)(12,13,17,14)(19,22,23,20)(25,26,29,31)(27,35,38,40)(28,34,30,33)(32,36,39,37);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,19)(13,22)(14,20)(15,21)(17,23)(18,24)(26,31)(27,36)(28,30)(32,35)(37,38)(39,40);(9,12,14,10,15,17,13,18)(11,19,20,16,21,23,22,24)(25,27,30,36,29,38,28,37)(26,32,34,40,31,39,33,35);(2,6)(5,7)(9,13,15,14)(10,17,18,12)(11,20,21,22)(16,19,24,23)(25,28,29,30)(26,33,31,34)(27,37,38,36)(32,35,39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=fi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 200396);