#142642

order	512
nilpotency class	4
isoclinism	isoclinic class of #142640
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	#25785	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 614 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=dh,e^c=eg,g^c=gi,h^c=hi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49838);