#187245

order	512
nilpotency class	3
isoclinism	isoclinic class of #187168
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62835	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2187 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:

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

Polycyclic presentation:

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

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, 94441);