#164341

order	512
nilpotency class	3
isoclinism	isoclinic class of #164127
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

987 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=gi,e^2=h,f^2=i,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dgi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 71537);