#112337

order	512
nilpotency class	3
isoclinism	isoclinic class of #112331
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	#35191	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19533);