#138683

order	512
nilpotency class	4
isoclinism	isoclinic class of #138564
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

328 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=h,d^2=ghi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,e^c=eh,f^c=fhi

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, 45879);