#138934

order	512
nilpotency class	4
isoclinism	isoclinic class of #138850
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

558 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,5,7,8)(4,6)(9,10,12,18,14,20,26,15,13,19,28,17,29,36,40,31)(11,21,22,34,25,37,27,33,24,32,23,38,30,35,39,16);(1,3,4,7)(2,5)(6,8)(10,15,19,31)(12,26)(14,29)(16,21,33,32)(17,20,18,36)(22,27)(23,39)(25,30)(28,40)(34,35,38,37);(9,11)(10,16)(12,27)(13,24)(14,30)(15,21)(17,35)(18,37)(19,33)(20,38)(22,26)(23,40)(25,29)(28,39)(31,32)(34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=ghi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=egh,f^c=fghi,h^c=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46130);