#234242

order	512
nilpotency class	3
isoclinism	isoclinic class of #234235
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1270 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)(3,6)(4,7)(5,9)(8,13)(10,15)(11,14)(12,16)(17,18,23,27)(19,28,20,31)(21,26,22,25)(24,30,32,29);(1,3)(2,6)(4,12)(5,10)(7,16)(8,14)(9,15)(11,13)(17,19,23,20)(18,24,27,32)(21,30,22,29)(25,31,26,28);(2,7)(4,13)(6,14)(8,9)(11,12)(15,16)(17,20,23,19)(18,24,27,32)(21,29,22,30)(25,31,26,28);(3,10)(6,15)(11,12)(14,16)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fi,e^c=eh,f^c=fi

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, 141438);