#264122

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

938 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)(5,6,9,12)(7,13,16,20)(8,17,14,11)(10,18,19,15)(21,22)(23,31)(24,27)(25,30)(26,29)(28,34)(32,35)(33,36);(5,7,14,18,9,16,8,15)(6,10,17,20,12,19,11,13)(22,27)(23,32)(25,26)(28,31)(33,34)(35,36);(2,4)(21,23,24,33)(22,28,27,35)(25,32,26,34)(29,36,30,31);(5,7,14,18,9,16,8,15)(6,10,17,20,12,19,11,13)(22,27)(28,35)(29,30)(31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171318);