#146843

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1770
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 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)(5,6,9,17,11,16,25,31,12,15,10,18,8,14,24,30)(7,19,21,32,22,27,34,35,23,13,26,29,20,28,33,36);(5,7,10,21,11,22,24,34,12,23,9,26,8,20,25,33)(6,13,18,29,16,28,30,36,15,19,17,32,14,27,31,35);(2,4)(6,14,15,16)(13,27,19,28)(17,30,18,31)(29,35,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54039);