#187674

order	512
nilpotency class	3
isoclinism	isoclinic class of #187667
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	#62748	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2245 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:

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

Polycyclic presentation:

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

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, 94870);