#289054

order	512
nilpotency class	3
isoclinism	isoclinic class of #289043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 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:

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

Polycyclic presentation:

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

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, 196250);