#334122

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
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

757 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 241318);