#502844

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

896 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410040);