#140193

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1924
prime radical	C₂³	#435
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

226 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

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

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, 47389);