#146852

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1924
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

486 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)(5,6,11,17,13,15,28,26,10,16,12,7,9,20,27,19)(8,14,23,31,24,30,36,35,25,32,29,18,22,34,33,21);(5,7,12,15,13,19,27,16,10,17,11,20,9,26,28,6)(8,21,23,14,24,31,36,30,25,35,29,32,22,18,33,34);(2,4)(5,8,9,22,10,25,13,24)(6,14,15,30,16,32,20,34)(7,18,19,21,17,31,26,35)(11,23,27,33,12,29,28,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ch,d^b=di,e^c=ei,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54048);