#456305

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
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

749 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:

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

Polycyclic presentation:

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

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, 363501);