#381065

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=f,e^2=i,h^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, 288261);