#441553

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

745 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348749);