#139786

order	512
nilpotency class	4
isoclinism	isoclinic class of #139783
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	#3486	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

694 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,c^b=ci,d^b=dgi,e^b=ehi,f^b=fgi,d^c=dh,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268477, #72268479
maximal normal subgroup of	#72268477, #72268479

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46982);