#138923

order	512
nilpotency class	4
isoclinism	isoclinic class of #138838
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

526 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,5,7,8)(4,6)(9,10,12,20,15,22,13,16,14,21,29,18,31,39,32,19)(11,23,24,17,27,35,25,37,26,38,30,36,33,34,28,40);(1,3,4,7)(2,5)(6,8)(10,16,21,19)(12,13)(15,31)(17,34,36,35)(18,22,20,39)(23,37,38,40)(24,25)(27,33)(28,30)(29,32);(9,11,14,26)(10,17,21,36)(12,28,29,25)(13,30,32,24)(15,33,31,27)(16,34,19,35)(18,38,20,23)(22,40,39,37)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=i,d^2=ghi,e^2=gh,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egh,f^b=fh,h^b=hi,d^c=dgh,e^c=eghi,f^c=fgh,h^c=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46119);