#441728

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=ef,d^2=ei,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, 348924);