#140189

order	512
nilpotency class	4
isoclinism	isoclinic class of #13156
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1936
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#440
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

245 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=d,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=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, 47385);