#131490

order	512
nilpotency class	4
isoclinism	isoclinic class of #131212
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×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

540 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gi,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,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, 38686);