#141267

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

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, 48463);