#187908

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

2059 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,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, 95104);