#145481

order	512
nilpotency class	4
isoclinism	isoclinic class of #145480
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
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 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,5)(4,7)(6,8)(9,10,15,21)(11,22,26,12)(13,31,35,27)(14,36,33,20)(16,24,38,23)(17,28,32,29)(18,39,19,34)(25,30,37,40);(3,8)(5,6)(9,11)(10,16)(12,27)(13,32)(14,37)(15,26)(17,35)(18,40)(19,30)(20,29)(21,38)(22,31)(23,34)(24,39)(25,33)(28,36);(2,5)(3,8)(6,7)(9,12,13,28,33,40,39,16,15,22,35,29,14,30,34,38)(10,17,18,26,36,24,27,25,21,32,19,11,20,23,31,37)

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,h^b=hi,e^c=egh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52677);