#435459

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1011 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)(3,4)(5,6,12,17,11,16,10,18)(7,19,22,30,24,13,23,33)(8,25,28,9,26,27,15,14)(20,34,32,31,35,36,29,21);(5,7,10,22,11,24,12,23)(6,13,18,33,16,19,17,30)(8,20,28,29,26,35,15,32)(9,21,27,36,14,31,25,34);(5,8,10,15,11,26,12,28)(6,14,17,25,16,9,18,27)(7,20,22,32,24,35,23,29)(13,21,30,36,19,31,33,34);(2,4)(5,9)(6,15)(7,21)(8,18)(10,27)(11,14)(12,25)(13,29)(16,28)(17,26)(19,32)(20,30)(22,36)(23,34)(24,31)(33,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,g^d=gi,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, 342655);