#264184

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1009 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)(5,6,10,14)(7,12,15,20)(8,17,16,11)(9,19,18,13)(21,22)(23,30)(24,28)(25,27)(26,32)(29,34)(31,36)(33,35);(5,7,9,16,10,15,18,8)(6,11,13,12,14,17,19,20)(21,23)(22,26)(24,32)(25,31)(27,35)(28,30)(29,36)(33,34);(2,4)(7,15)(8,16)(11,17)(12,20)(22,27)(23,31)(25,34)(28,29)(32,33)(35,36);(5,8,18,15,10,16,9,7)(6,12,19,11,14,20,13,17)(22,28)(26,30)(27,29)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,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,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171380);