#149085

order	512
nilpotency class	4
isoclinism	isoclinic class of #140211
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1936
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,g^b=ghi,h^b=hi,f^c=fi,g^c=gi

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, 56281);