#446151

order	512
nilpotency class	3
isoclinism	isoclinic class of #423425
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

996 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=fgi,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353347);