#6357722

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357716
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2023 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^c=eh,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264918);