#140701

order	512
nilpotency class	4
isoclinism	isoclinic class of #140675
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1964
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9654
prime radical	C₂⁴	#131
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

351 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^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=dg,f^b=fgi,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47897);