#140690

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	#1466
prime radical	C₂×#525	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

551 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 88 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)(5,6,10,17,11,7,18,20,12,15,29,30,9,16,13,27)(8,21,24,36,25,14,31,32,26,19,28,35,23,22,34,33);(5,7,10,27,9,6,13,30,12,16,29,20,11,15,18,17)(8,22,24,32,23,19,34,36,26,14,28,33,25,21,31,35);(1,2,3,4)(5,8,9,23,12,26,11,25)(6,14,7,19,15,22,16,21)(10,28,13,31,29,24,18,34)(17,33,20,36,30,32,27,35)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eg,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dg,f^b=fg,g^b=gi,f^c=fi,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, 47886);