#111395

order	512
nilpotency class	3
isoclinism	isoclinic class of #111390
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

834 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18591);