#142629

order	512
nilpotency class	4
isoclinism	isoclinic class of #142624
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	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

655 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 655 conjugacy classes of subgroups — too many to draw. Only its 76 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)(9,10,14,19)(11,20,23,38)(12,15,27,31)(13,28,26,18)(16,32,34,21)(17,35,30,24)(22,39,29,36)(25,37,40,33);(5,8)(10,15)(12,24)(13,26)(16,20)(17,18)(19,31)(21,25)(22,29)(27,35)(28,30)(32,40)(33,39)(34,38)(36,37);(1,3)(2,5)(4,7)(6,8)(9,11,14,23)(10,16,19,34)(12,25,27,40)(13,29,26,22)(15,20,31,38)(17,36,30,39)(18,37,28,33)(21,35,32,24)

Polycyclic presentation:

2^9|a^2=i,c^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=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=dh,e^c=eg,g^c=gi,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49825);