#140677

order	512
nilpotency class	4
isoclinism	isoclinic class of #129430
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1464
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

407 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 84 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)(5,6,11,17,9,16,20,33,12,18,27,14,10,15,8,25)(7,19,23,35,22,32,36,34,24,30,21,29,28,13,26,31);(5,7,20,36,9,22,27,21,12,24,8,26,10,28,11,23)(6,13,25,35,16,19,17,34,18,32,33,29,15,30,14,31);(1,2,3,4)(5,8,10,27,12,20,9,11)(6,14,16,25,18,17,15,33)(7,21,22,26,24,23,28,36)(13,29,30,34,32,35,19,31)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch,d^b=dgi,e^b=ei,f^b=fgi,g^b=gi,d^c=di,f^d=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, 47873);