#250524

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
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

1132 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)(3,8)(4,12)(5,7)(6,10)(9,11)(13,15)(14,16)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32);(2,7)(3,5)(8,10)(9,12)(11,14)(15,16)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,31)(28,32);(1,3,6,5)(2,8,10,7)(4,11,13,14)(9,15,16,12)(17,19)(18,23)(20,29)(21,26)(22,28)(24,32)(25,31)(27,30);(1,4,6,13)(2,9,10,16)(3,11,5,14)(7,12,8,15)(17,18)(19,23)(20,30)(21,24)(22,25)(26,32)(27,29)(28,31)

Polycyclic presentation:

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

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, 157720);