#113357

order	512
nilpotency class	3
isoclinism	isoclinic class of #113356
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

943 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20553);