#334673

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	#18674	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

703 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:

(1,2,6,8,5,10,4,9)(3,11,12,7,14,16,13,15)(17,18)(19,24)(20,29)(21,31)(22,27)(23,28)(25,30)(26,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(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);(2,8)(3,12)(4,6)(9,10)(11,16)(13,14)(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=i,c^2=i,d^2=i,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, 241869);