#131990

order	512
nilpotency class	4
isoclinism	isoclinic class of #131988
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

854 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227716, #72227719, #72227720
maximal normal subgroup of	#72227716, #72227719, #72227720

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39186);