#441630

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1130 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=efgi,c^2=ef,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348826);