#112813

order	512
nilpotency class	3
isoclinism	isoclinic class of #112799
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	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1072 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:

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

Polycyclic presentation:

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

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, 20009);