#111802

order	512
nilpotency class	3
isoclinism	isoclinic class of #111793
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

887 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,f^2=h,g^2=i,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, 18998);