#132228

order	512
nilpotency class	4
isoclinism	isoclinic class of #132185
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#553 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

661 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 661 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,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, 39424);