#162963

order	512
nilpotency class	3
isoclinism	isoclinic class of #162948
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=g,d^2=g,e^2=h,f^2=i,b^a=be,d^a=di,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, 70159);