#141906

order	512
nilpotency class	4
isoclinism	isoclinic class of #13291
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1993
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 287 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)(9,10,15,20,16,21,14,19)(11,22,24,32,25,31,18,13)(12,26,29,36,30,35,33,17)(23,28,34,38,39,40,27,37);(1,3,4,8,5,2,6,7)(9,11,13,21,14,24,31,19,16,25,32,10,15,18,22,20)(12,27,40,35,29,39,37,36,30,34,28,26,33,23,38,17);(9,12)(10,17,21,36)(11,23,25,39)(13,28)(14,29)(15,33)(16,30)(18,34,24,27)(19,26,20,35)(22,37)(31,38)(32,40)

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=egi,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,e^b=ei,f^b=fi,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, 49102);