#234201

order	512
nilpotency class	3
isoclinism	isoclinic class of #234178
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1266 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)(3,11)(4,9)(5,8)(6,10)(7,13)(12,16)(14,15)(17,18)(19,23)(20,28)(21,30)(22,26)(24,29)(25,31)(27,32);(1,3,6,14)(2,7,10,16)(4,12,5,13)(8,15,9,11)(17,19,22,20)(18,23,26,28)(21,29,31,27)(24,25,32,30);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,20,22,19)(18,24,26,32)(21,27,31,29)(23,30,28,25);(1,4,6,5)(2,9,10,8)(3,12,14,13)(7,11,16,15)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^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,e^b=eh,f^b=fi,d^c=dh,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141397);