#441744

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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

918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=fgi,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, 348940);