#293185

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

738 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)(9,10,14,18)(11,15,16,24)(12,20,23,19)(13,21,22,17)(28,36)(30,38)(32,37)(35,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,19)(14,16)(17,23)(18,24)(20,22)(25,26)(27,33)(28,37)(29,31)(30,35)(32,36)(34,39)(38,40);(9,12,22,24,14,23,13,15)(10,16,21,19,18,11,17,20)(25,27,29,34)(26,30)(28,35,36,40)(31,38)(32,33)(37,39);(2,6)(5,7)(11,16)(12,23)(15,24)(19,20)(27,34)(30,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,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, 200381);