#243097

order	512
nilpotency class	3
isoclinism	isoclinic class of #243097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

617 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:

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

Polycyclic presentation:

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

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, 150293);