#440776

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=ei,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 347972);