#144038

order	512
nilpotency class	4
isoclinism	isoclinic class of #143959
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

399 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,5,8,6)(9,10)(11,16)(12,15)(13,28)(14,20)(17,24)(18,36)(19,26)(21,35)(22,38)(23,34)(25,30)(27,31)(29,32)(33,40)(37,39);(10,15)(11,21)(12,24)(13,26)(16,32)(17,19)(18,28)(20,31)(22,40)(23,39)(25,29)(27,36)(30,33)(34,38);(2,5)(3,8)(6,7)(9,11,14,23)(10,16,20,34)(12,25,27,40)(13,29,26,22)(15,30,31,33)(17,35,18,37)(19,38,28,32)(21,36,39,24)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,c^b=cf,d^b=dg,f^b=fg,g^b=gi,d^c=dg,e^c=eh,f^c=fg,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51234);