#334458

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	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

749 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,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, 241654);