#264567

order	512
nilpotency class	3
isoclinism	isoclinic class of #264555
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

876 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)(7,15)(8,11)(9,18)(10,14)(12,16)(13,19)(17,20)(21,22,25,29)(23,31,34,27)(24,30,26,28)(32,36,33,35);(5,7,9,8,10,16,19,17)(6,11,13,15,14,20,18,12)(21,23)(22,27)(24,33)(25,34)(26,32)(28,35)(29,31)(30,36);(2,4)(7,16)(8,17)(11,20)(12,15)(22,28)(23,32)(24,26)(29,30)(33,34)(35,36);(5,8,9,16,10,17,19,7)(6,12,13,11,14,15,18,20)(21,24,25,26)(22,28,29,30)(23,33,34,32)(27,35,31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171763);