#282100

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

700 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)(21,22,27,29,26,24,25,30)(23,28,34,35,33,31,32,36);(2,4)(5,7,11,13,10,6,9,14)(8,17,18,12,19,16,20,15)(21,23,27,34,26,33,25,32)(22,28,29,35,24,31,30,36);(5,8)(6,12)(7,15)(9,18)(10,19)(11,20)(13,16)(14,17)(21,24,26,22)(23,31,33,28)(25,30,27,29)(32,36,34,35);(2,4)(5,9,10,11)(6,13,7,14)(8,18,19,20)(12,16,15,17)(21,25,26,27)(22,29,24,30)(23,32,33,34)(28,35,31,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=i,d^2=h,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, 189296);