#138927

order	512
nilpotency class	4
isoclinism	isoclinic class of #138844
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	#33520	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 555 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)(9,10,12,18,26,37,29,33,15,20,27,32,14,19,13,16)(11,21,22,36,30,31,23,35,25,39,28,17,24,34,40,38);(1,3,4,7)(2,5)(6,8)(10,16)(12,13)(14,26)(17,31)(18,19)(20,33)(21,38)(22,40)(23,28)(24,30)(27,29)(32,37)(34,36)(35,39);(9,11,15,25)(10,17,20,36)(12,23,27,40)(13,28,29,22)(14,30,26,24)(16,31,33,34)(18,21,32,39)(19,38,37,35)

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,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=eghi,f^c=fghi,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, 46123);