#140884

order	512
nilpotency class	4
isoclinism	isoclinic class of #140881
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 393 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ei,d^2=gh,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=fh,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, 48080);