#148911

order	512
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1947
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=hi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,f^b=fi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56107);