#138918

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	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

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

normal subgroups only

This group has 528 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=gh,f^2=gh,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, 46114);