#187906

order	512
nilpotency class	3
isoclinism	isoclinic class of #187866
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	#62761	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2082 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,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, 95102);