#243010

order	512
nilpotency class	3
isoclinism	isoclinic class of #243002
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

654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=di,f^c=fh,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, 150206);