#334786

order	512
nilpotency class	3
isoclinism	isoclinic class of #334695
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

626 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=h,c^2=i,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=defh,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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 241982);