#498134

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

800 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)(16,28)(17,32)(18,29)(19,33)(20,35)(23,26)(25,27)(30,40)(31,39)(34,37)(36,38);(5,7,8,6)(9,10,14,18)(11,16,21,31)(12,23,25,32)(13,26,24,17)(15,27,29,22)(19,34,36,40)(20,37,35,30)(28,38,39,33);(2,4)(9,11,14,21)(10,16,18,31)(12,19,25,36)(13,20,24,35)(15,28,29,39)(17,30,26,37)(22,33,27,38)(23,34,32,40);(10,15)(12,22)(13,24)(16,28)(17,32)(18,29)(19,33)(20,35)(23,26)(25,27)(30,40)(31,39)(34,37)(36,38)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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, 405330);