#497843

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×D₄	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

680 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)(11,16)(12,22)(13,26)(14,29)(15,19)(17,23)(18,27)(20,34)(21,33)(24,40)(25,37)(28,31)(30,35)(32,39)(36,38);(1,2)(3,4)(9,11,15,14)(10,16,19,29)(12,20,25,24)(13,21,28,27)(17,30,32,36)(18,26,33,31)(22,34,37,40)(23,35,39,38);(9,12,21,35,14,24,13,23,15,25,27,38,11,20,28,39)(10,17,31,40,16,30,18,22,19,32,26,34,29,36,33,37);(2,4)

Polycyclic presentation:

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

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, 405039);