#234268

order	512
nilpotency class	3
isoclinism	isoclinic class of #234257
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

1291 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,16)(14,15)(17,18)(19,28)(20,31)(21,25)(22,26)(23,27)(24,29)(30,32);(1,3,6,12)(2,7,9,15)(4,11,5,14)(8,16,13,10)(17,19,23,20)(18,24,27,32)(21,30,22,29)(25,28,26,31);(4,5)(7,15)(8,13)(11,14)(17,20,23,19)(18,24,27,32)(21,29,22,30)(25,28,26,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,16,15,10)(17,21,23,22)(18,25,27,26)(19,29,20,30)(24,31,32,28)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141464);