#187911

order	512
nilpotency class	3
isoclinism	isoclinic class of #187872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,d^c=dh,e^c=ehi,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, 95107);