#497378

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	C₂×#3280	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,f^c=fh,h^c=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, 404574);