#111794

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

931 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=h,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, 18990);