#339856

order	512
nilpotency class	3
isoclinism	isoclinic class of #339847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

576 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)(4,6)(5,9)(7,14)(8,13)(11,16)(12,15)(18,24)(19,26)(20,28)(21,22)(25,29)(27,32);(2,6)(3,11)(4,13)(8,9)(10,15)(12,14)(17,18)(19,23)(20,29)(21,26)(22,24)(25,30)(27,31)(28,32);(1,2,5,9)(3,7,12,16)(4,8,13,6)(10,11,15,14)(17,19,23,18)(20,27,28,25)(21,26,22,24)(29,30,32,31);(1,3,4,11,5,12,13,14)(2,7,8,15,9,16,6,10)(17,20,21,30,23,28,22,31)(18,25,24,29,19,27,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247052);