#130640

order	512
nilpotency class	4
isoclinism	isoclinic class of #130482
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#556 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 652 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)(11,21,24,34)(12,25,27,33)(13,19,14,18)(15,20)(16,23)(17,26)(22,36)(28,32)(29,31)(30,39)(35,37,40,38);(1,2)(3,7)(4,6)(5,8)(9,11,15,24)(10,16)(12,22,27,39)(13,23,14,29)(17,30)(18,21)(19,34)(20,31)(25,40)(26,37,28,38)(32,36)(33,35);(9,12)(10,17,20,32)(11,22)(13,28)(14,26)(15,27)(16,30,31,36)(18,33,19,25)(21,35,34,40)(23,38)(24,39)(29,37)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=g,d^2=hi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fghi,d^c=di,f^c=fgi,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37836);