#292990

order	512
nilpotency class	3
isoclinism	isoclinic class of #271259
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

755 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)(11,20,22,16)(12,21,23,24)(13,19,15,17)(25,26,29,32)(27,30,34,39)(28,36,37,31)(33,40,35,38);(1,3,4,8)(2,5,6,7)(9,11,14,22)(10,16,18,20)(12,19,23,17)(13,24,15,21)(25,27)(26,30)(28,33)(29,34)(31,38)(32,39)(35,37)(36,40);(9,12,13,20,14,23,15,16)(10,11,17,24,18,22,19,21)(27,33,34,35)(30,38,39,40);(2,6)(5,7)(9,13,14,15)(10,17,18,19)(11,21,22,24)(12,16,23,20)

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=fh,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=dh

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, 200186);