#111804

order	512
nilpotency class	3
isoclinism	isoclinic class of #111793
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1062 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 19000);