#281560

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

714 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)(5,6,8,12,10,14,9,13)(7,15,18,20,19,16,17,11)(21,22,24,29,25,28,31,27)(23,30,32,35,33,26,34,36);(2,4)(5,6,8,12,10,14,9,13)(7,16,18,11,19,15,17,20)(21,23,31,34,25,33,24,32)(22,26,27,35,28,30,29,36);(5,7,10,19)(6,11,14,20)(8,17,9,18)(12,16,13,15)(21,22)(23,30)(24,27)(25,28)(26,33)(29,31)(32,36)(34,35);(2,4)(5,8,10,9)(6,12,14,13)(7,17,19,18)(11,16,20,15)(21,24,25,31)(22,27,28,29)(23,32,33,34)(26,35,30,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=hi,d^2=h,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, 188756);