#6464098

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=hi,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,e^c=ei

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, 6371294);