#340052

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

696 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=de,e^b=eh,f^c=fi,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, 247248);