#161864

order	512
nilpotency class	3
isoclinism	isoclinic class of #161854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,8,6,3,4,7)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23)(25,26)(27,33)(28,37)(29,34)(30,39)(31,35)(32,36)(38,40);(1,3,6,2)(4,7,5,8)(9,11,13,19)(10,14,15,22)(12,20,21,18)(16,24,23,17)(25,27,31,28)(26,32,35,40)(29,36,30,38)(33,34,37,39);(2,3)(7,8)(10,15)(14,22)(16,23)(17,24)(25,28,31,27)(26,33,35,37)(29,36,30,38)(32,34,40,39);(25,29,31,30)(26,34,35,39)(27,36,28,38)(32,37,40,33)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=g,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,c^b=ci,e^b=eh,f^b=fgh,d^c=dg

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, 69060);