#142845

order	512
nilpotency class	4
isoclinism	isoclinic class of #142798
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#3449	C₂²
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

587 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50041);