#236653

order	512
nilpotency class	3
isoclinism	isoclinic class of #236644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
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

1220 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:

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

Polycyclic presentation:

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

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, 143849);