#381064

order	512
nilpotency class	3
isoclinism	isoclinic class of #380391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2331
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14284
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

677 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)(3,11,14,27)(4,15)(5,9)(6,10)(7,13,21,32)(8,18)(12,28,31,26)(16,24)(17,23)(19,25)(20,30,22,29);(1,3)(2,7,10,21)(4,12)(5,13)(6,14)(8,22,15,20)(9,11,25,27)(16,29)(17,30)(18,31)(19,32)(23,28,24,26);(1,4)(2,8,10,15)(3,12)(5,16)(6,18)(7,20,21,22)(9,23,25,24)(11,26,27,28)(13,29)(14,31)(17,19)(30,32);(4,16,18,17)(8,23,15,24)(12,29,31,30)(20,26,22,28)

Polycyclic presentation:

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

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, 288260);