#140635

order	512
nilpotency class	4
isoclinism	isoclinic class of #13229
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
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×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 383 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)(9,10,15,19,16,20,14,24)(11,21,18,22,26,13,25,32)(12,27,30,34,31,37,29,17)(23,36,33,28,39,40,38,35);(1,3,4,8,5,2,6,7)(9,11,22,10,16,26,32,20)(12,23,36,17,31,39,40,34)(13,19,15,18,21,24,14,25)(27,30,33,35,37,29,38,28);(9,12,15,30,16,31,14,29)(10,17,24,37,20,34,19,27)(11,23,18,33,26,39,25,38)(13,28,22,36,21,35,32,40)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=ei,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,f^b=fgi,g^b=gi,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, 47831);