#470718

order	512
nilpotency class	3
isoclinism	isoclinic class of #460462
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2499
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16696
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377914);