#343111

order	512
nilpotency class	3
isoclinism	isoclinic class of #343099
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	H(3,𝔽₄)
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

607 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=h,c^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,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, 250307);