#334413

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	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

716 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=i,c^2=h,d^2=i,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, 241609);