#334534

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

708 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=h,c^2=hi,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, 241730);