#130631

order	512
nilpotency class	4
isoclinism	isoclinic class of #130498
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#554 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

640 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 84 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)(9,10,13,18,14,19,27,34)(11,20,23,35,21,15,26,30)(12,17,24,33,25,16,28,32)(22,31,37,40,36,29,38,39);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15)(12,22,25,36)(13,26,27,23)(16,31)(17,29)(18,30)(19,20)(24,38,28,37)(32,40)(33,39)(34,35);(10,16,19,17)(11,21)(15,29,20,31)(18,33,34,32)(22,36)(23,26)(30,40,35,39)(37,38)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gi,d^2=hi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,f^b=fghi,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, 37827);