#282086

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=i,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 189282);