#211287

order	512
nilpotency class	3
isoclinism	isoclinic class of #19384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

721 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,f^c=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118483);