#234171

order	512
nilpotency class	3
isoclinism	isoclinic class of #234142
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	#35293	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1406 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:

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

Polycyclic presentation:

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

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, 141367);