#137733

order	512
nilpotency class	4
isoclinism	isoclinic class of #13096
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,e^b=ei,f^b=fgi,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44929);