#243143

order	512
nilpotency class	3
isoclinism	isoclinic class of #243142
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

599 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150339);