#391148

order	512
nilpotency class	3
isoclinism	isoclinic class of #32696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=eh,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298344);