#115125

order	512
nilpotency class	3
isoclinism	isoclinic class of #115124
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

913 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:

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

Polycyclic presentation:

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

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, 22321);