#131623

order	512
nilpotency class	4
isoclinism	isoclinic class of #131240
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

700 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,e^c=ei,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38819);