#140932

order	512
nilpotency class	4
isoclinism	isoclinic class of #140881
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#33179	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

497 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 102 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)(3,6)(7,8)(9,10,15,19,16,20,14,30)(11,13,18,21,24,31,23,32)(12,25,28,34,29,38,27,17)(22,35,33,39,37,40,36,26);(1,2,6,8,4,7,3,5)(9,11,21,10)(12,22,35,17)(13,30,15,18)(14,23,31,19)(16,24,32,20)(25,27,36,39)(26,38,28,33)(29,37,40,34);(9,12,15,28,16,29,14,27)(10,17,30,38,20,34,19,25)(11,22,18,33,24,37,23,36)(13,26,32,40,31,39,21,35)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=ei,d^2=ghi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48128);