#498135

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

814 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)(10,16)(11,15)(13,25)(17,32)(18,22)(19,31)(20,30)(21,28)(23,27)(26,29)(33,40)(34,38)(35,37)(36,39);(5,7,8,6)(9,10,14,19)(11,17,15,29)(12,22,24,39)(13,26,28,32)(16,21,31,25)(18,34,36,27)(20,35,30,40)(23,37,38,33);(2,4)(9,11,14,15)(10,17,19,29)(12,20,24,30)(13,21,28,25)(16,26,31,32)(18,33,36,37)(22,35,39,40)(23,27,38,34);(9,12,14,24)(10,18,19,36)(11,20,15,30)(13,27,28,34)(16,22,31,39)(17,33,29,37)(21,38,25,23)(26,35,32,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^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, 405331);