#140746

order	512
nilpotency class	4
isoclinism	isoclinic class of #13247
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 332 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cg,e^b=ei,f^c=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, 47942);