#334690

order	512
nilpotency class	3
isoclinism	isoclinic class of #334567
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

637 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 241886);