#138921

order	512
nilpotency class	4
isoclinism	isoclinic class of #138838
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

539 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,5,7,8)(4,6)(9,10,14,19,16,21,13,31,15,20,12,28,30,35,29,17)(11,22,25,18,27,34,24,37,26,39,23,36,33,40,32,38);(1,3,4,7)(2,5)(6,8)(10,17,20,31)(12,29)(13,14)(16,30)(18,34,36,40)(19,21,28,35)(22,38,39,37)(23,32)(24,25)(27,33);(9,11,15,26)(10,18,20,36)(12,24,14,32)(13,23,29,25)(16,33,30,27)(17,34,31,40)(19,22,28,39)(21,38,35,37)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=ghi,e^2=gh,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egh,f^b=fh,h^b=hi,d^c=dgh,e^c=eghi,f^c=fgh,h^c=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46117);