#498352

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×#466	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

678 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)(9,10,15,20,31,38,29,33,16,21,14,19,12,18,30,13)(11,22,26,37,23,40,34,17,28,32,25,36,27,39,35,24);(5,7,8,6)(9,11,16,28)(10,17,21,24)(12,23,31,27)(13,32,33,22)(14,34,15,35)(18,36,38,37)(19,39,20,40)(25,30,26,29);(2,4)(9,12,16,31)(10,18,21,38)(11,23,28,27)(13,19,33,20)(14,29,15,30)(17,36,24,37)(22,40,32,39)(25,35,26,34);(9,13,30,18,12,19,14,21,16,33,29,38,31,20,15,10)(11,24,35,39,27,36,25,32,28,17,34,40,23,37,26,22)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=ehi,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=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, 405548);