#498130

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

828 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)(3,4)(5,6,8,7)(10,15)(12,22)(13,24)(16,27)(17,31)(18,28)(19,32)(20,34)(23,26)(25,37)(29,39)(30,38)(33,36)(35,40);(5,7,8,6)(9,10)(11,16)(12,23)(13,26)(14,18)(15,22)(17,24)(19,33)(20,36)(21,30)(25,31)(27,32)(28,37)(29,34)(35,39)(38,40);(2,4)(9,11,14,21)(10,16,18,30)(12,19,25,35)(13,20,24,34)(15,27,28,38)(17,29,26,36)(22,32,37,40)(23,33,31,39);(10,15)(12,22)(13,24)(16,27)(17,31)(18,28)(19,32)(20,34)(23,26)(25,37)(29,39)(30,38)(33,36)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=deh,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405326);