#6464093

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371289);