#340414

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

635 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)(7,13,16,17)(8,9,18,19)(11,20,14,15)(21,22)(23,30)(24,29)(25,27)(26,31)(28,33)(32,35)(34,36);(1,2)(3,4)(5,7,10,16)(6,11,12,14)(8,17,18,13)(9,20,19,15)(21,23,24,31)(22,26,29,30)(25,32,33,36)(27,34,28,35);(2,4)(6,12)(8,18)(11,14)(13,17)(22,27)(25,33)(26,34)(28,29)(30,35)(32,36);(5,8,19,6,10,18,9,12)(7,14,20,13,16,11,15,17)(22,28)(25,33)(26,35)(27,29)(30,34)(32,36)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=ehi,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=dei,e^b=eh,d^c=dh,f^c=fi,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, 247610);