#161862

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	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

748 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)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(25,26)(27,33,35,40)(28,32)(29,31)(30,34,38,36)(37,39);(1,3,6,2)(4,7,5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,35)(26,30,31,38)(28,36,37,34)(32,40,39,33);(2,3)(7,8)(12,20)(15,22)(18,24)(21,23)(26,31)(30,38)(32,39)(33,40);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)

Polycyclic presentation:

2^9|a^2=f,b^2=h,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, 69058);