#281617

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
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

706 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)(5,6,9,14,11,16,10,15)(7,12,17,13,8,19,18,20)(21,22,27,29,26,24,25,30)(23,31,34,36,33,28,32,35);(2,4)(5,7,9,17,11,8,10,18)(6,12,14,13,16,19,15,20)(21,23,27,34,26,33,25,32)(22,28,29,35,24,31,30,36);(5,8,11,7)(6,13,16,20)(9,17,10,18)(12,15,19,14)(21,24,26,22)(23,28,33,31)(25,30,27,29)(32,35,34,36);(2,4)(21,25,26,27)(22,29,24,30)(23,32,33,34)(28,35,31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188813);