#137582

order	512
nilpotency class	4
isoclinism	isoclinic class of #137582
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9064
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 ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 439 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) = 32
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=g,e^2=ghi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fgi,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219575, #72219599, #72219600
maximal normal subgroup of	#72219575, #72219599, #72219600
maximal quotient of	#72226846

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44778);