#453885

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16199
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

557 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361081);