#343399

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,11,14,10,7,8,12)(9,17,16,15,19,20,18,13)(21,22,23,29,25,30,26,27)(24,32,34,28,33,35,31,36);(1,2)(3,4)(5,7,10,6)(8,12,11,14)(9,17,19,20)(13,18,15,16)(21,23,25,26)(22,27,30,29)(24,31,33,34)(28,35,36,32);(2,4)(5,8,10,11)(6,12,7,14)(9,16,19,18)(13,17,15,20)(21,24,25,33)(22,28,30,36)(23,31,26,34)(27,35,29,32);(5,9,10,19)(6,13,7,15)(8,16,11,18)(12,17,14,20)(22,29)(23,26)(27,30)(28,32)(31,34)(35,36)

Polycyclic presentation:

2^9|a^2=efh,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,e^b=eh,f^c=fi,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, 250595);