#497445

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#508	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

1028 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)(5,6,7,8)(10,16)(12,24)(13,26)(14,28)(17,30)(18,25)(19,31)(20,34)(21,36)(22,38)(27,29)(32,35)(33,40)(37,39);(9,10,15,18)(11,17,23,32)(12,25,26,16)(13,27,24,31)(14,29,28,19)(20,35,36,30)(21,37,34,40)(22,39,38,33);(2,4)(9,11,15,23)(10,17,18,32)(12,20,26,36)(13,21,24,34)(14,22,28,38)(16,30,25,35)(19,33,29,39)(27,37,31,40);(9,12)(10,18)(11,20)(13,28)(14,24)(15,26)(16,31)(17,32)(21,38)(22,34)(23,36)(25,27)(30,40)(35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,h^a=hi,d^b=de,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, 404641);