#162404

order	512
nilpotency class	3
isoclinism	isoclinic class of #162342
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×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

705 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)(11,19,21,23)(12,16)(13,17)(14,18)(15,20,24,22)(25,26)(27,33)(28,36)(29,31)(30,34)(32,38)(35,40)(37,39);(1,3,6,2)(4,7,5,8)(9,11,14,21)(10,15,18,24)(12,20,13,22)(16,19,17,23)(25,27,29,35)(26,30,31,39)(28,37,38,34)(32,40,36,33);(1,4,6,5)(2,7,3,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,19,24,23)(25,26)(27,30)(28,32)(29,31)(33,34)(35,39)(36,38)(37,40);(1,4,6,5)(2,8,3,7)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,23,24,19)(26,31)(30,39)(32,36)(33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gh,d^2=gh,e^2=h,f^2=i,b^a=be,d^a=dh,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, 69600);