#240657

order	512
nilpotency class	3
isoclinism	isoclinic class of #240653
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

847 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,e^2=h,f^2=i,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,d^c=dhi,f^c=fhi,g^c=gh

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, 147853);