#110208

order	512
nilpotency class	3
isoclinism	isoclinic class of #110191
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

746 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 194 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei,f^b=fi,d^c=di

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, 17404);