#140406

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1948
prime radical	C₂³	#428
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 232 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=ei,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch,d^b=dg,f^b=fi,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47602);