#333745

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
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	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

773 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:

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

Polycyclic presentation:

2^9|a^2=ehi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240941);