#112056

order	512
nilpotency class	3
isoclinism	isoclinic class of #111226
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	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

904 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)(3,7)(6,8)(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);(1,2,4,6)(3,8,7,5)(9,11,14,22)(10,15,17,19)(12,20,25,38)(13,27,29,21)(16,32,33,30)(18,34,26,31)(23,40,35,36)(24,39,28,37);(1,2)(3,8)(4,6)(5,7)(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,b^2=h,c^2=hi,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=di,e^c=eh

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, 19252);