#113126

order	512
nilpotency class	3
isoclinism	isoclinic class of #113119
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

897 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)(9,10,14,20,15,18,13,19)(11,21,24,32,25,16,23,33)(12,26,29,35,30,17,28,36)(22,31,38,40,39,37,27,34);(1,2)(3,7)(4,6)(5,8)(9,11,15,25)(10,16,18,21)(12,27,30,38)(13,24,14,23)(17,34,26,40)(19,33,20,32)(22,29,39,28)(31,36,37,35);(3,8)(5,7)(9,12,15,30)(10,17,18,26)(11,22,25,39)(13,29,14,28)(16,31,21,37)(19,36,20,35)(23,38,24,27)(32,40,33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20322);