#131255

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	C₂×C₄×D₄	C₂³
prime radical	C₂×#120	D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 78 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)(9,10,14,19,15,18,13,28)(11,20,22,32,23,16,29,33)(12,24,26,35,27,36,30,17)(21,37,39,34,25,40,38,31);(1,2)(3,7)(4,6)(5,8)(9,11,15,23)(10,16,18,20)(12,25,27,21)(13,29,14,22)(17,34,35,31)(19,32,28,33)(24,37,36,40)(26,38,30,39);(9,12,15,27)(10,17)(11,21,23,25)(13,30,14,26)(16,31)(18,35)(19,36)(20,34)(22,39,29,38)(24,28)(32,37)(33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^c=ei,f^c=fg,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, 38451);