#112167

order	512
nilpotency class	3
isoclinism	isoclinic class of #112162
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	#60803	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

908 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19363);