#142829

order	512
nilpotency class	4
isoclinism	isoclinic class of #142828
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#33520	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

658 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50025);