#132280

order	512
nilpotency class	4
isoclinism	isoclinic class of #132279
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2353 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

526 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 76 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)(11,21,25,32)(12,26,29,36)(13,19,14,18)(15,20)(16,23)(17,28)(22,31)(24,33)(27,34)(30,35)(37,39,40,38);(1,2)(3,7)(4,6)(5,8)(9,11,15,25)(10,16,20,33)(12,27,29,22)(13,24,14,23)(17,34,35,31)(18,32,19,21)(26,40,36,37)(28,38,30,39);(3,8)(5,7)(9,12)(10,17,20,35)(11,22)(13,30)(14,28)(15,29)(16,31,33,34)(18,36,19,26)(21,37,32,40)(23,39)(24,38)(25,27)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,e^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=eh,f^b=fgh,d^c=dh,f^c=fgi,f^d=fi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39476);