#498557

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×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 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:

(2,4)(5,6,8,7)(9,10,12,18,24,30,36,26,13,16,25,28,14,19,21,15)(11,17,20,31,33,40,37,38,22,29,34,39,23,32,35,27);(5,7,8,6)(10,15)(11,20)(12,21)(14,24)(16,26)(18,19)(22,34)(23,37)(25,36)(27,31)(28,30)(32,40)(33,35)(38,39);(1,2,3,4)(10,16)(15,26)(17,29)(18,28)(19,30)(27,38)(31,39)(32,40);(9,11,21,35,14,23,25,34,13,22,36,37,24,33,12,20)(10,17,15,27,19,32,28,39,16,29,26,38,30,40,18,31)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fi,d^2=eh,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, 405753);