#6357762

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

1872 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=gh,d^2=h,e^2=i,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, 6264958);