#497883

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×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 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)(5,6,7,8)(9,10)(11,19)(12,24)(13,16)(14,28)(15,21)(17,26)(18,25)(20,33)(22,36)(23,40)(27,35)(29,34)(30,38)(31,37)(32,39);(9,11,13,22)(10,15,16,30)(12,20,27,37)(14,29,26,23)(17,33,28,31)(18,34,32,40)(19,35,36,24)(21,39,38,25);(1,2,3,4)(10,16)(15,30)(17,28)(18,32)(19,36)(24,35)(31,33)(34,40);(10,17)(11,20)(12,25)(14,26)(15,19)(16,28)(18,32)(21,23)(22,37)(27,39)(29,38)(30,36)(31,40)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fi,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, 405079);