#250366

order	512
nilpotency class	3
isoclinism	isoclinic class of #250327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1055 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,f^c=fh,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157562);