#334566

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
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	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

754 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=hi,c^2=hi,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=dei,e^b=eh,f^c=fi,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, 241762);