#229531

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

703 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)(5,6,8,13,9,11,17,10)(7,14,15,18,16,12,19,20)(21,22)(23,30)(24,28)(25,27)(26,31)(29,33)(32,34)(35,36);(1,3,4,2)(6,10)(7,15)(8,17)(11,13)(16,19)(18,20)(21,23,24,31)(22,26,28,30)(25,32,33,36)(27,35,29,34);(6,11)(7,16)(10,13)(15,19)(22,27)(23,31)(25,33)(26,34)(28,29)(30,35);(5,7,17,19,9,16,8,15)(6,12,10,18,11,14,13,20)(23,31)(26,30)(32,36)(34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=dh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136727);