#140212

order	512
nilpotency class	4
isoclinism	isoclinic class of #140211
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1936
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

259 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,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=cgh,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.

Relationships

maximal subgroup of	#72320774, #72321061, #72321125
maximal normal subgroup of	#72320774, #72321061, #72321125

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47408);