#240815

order	512
nilpotency class	3
isoclinism	isoclinic class of #240803
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
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

936 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)(3,7)(4,9)(5,13)(6,10)(8,11)(12,16)(14,15)(18,22)(19,25)(20,28)(23,30)(24,29)(27,31);(1,3)(2,7)(4,11)(5,14)(6,12)(8,9)(10,16)(13,15)(17,18,21,24)(19,26,27,32)(20,29,28,22)(23,31,30,25);(1,3)(2,8)(4,11)(5,14)(6,12)(7,13)(9,16)(10,15)(17,19,20,25,21,27,28,31)(18,23,22,26,24,30,29,32);(1,4,6,5)(2,9,10,13)(3,11,12,14)(7,8,16,15)(18,24)(22,29)(23,30)(26,32)

Polycyclic presentation:

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

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, 148011);