#6357714

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357668
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

1944 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)(19,29)(20,23)(21,26)(22,27)(24,28)(25,30)(31,32);(1,2,8,16,6,11,5,7)(3,9,13,15,4,10,12,14)(17,19,23,31,24,26,22,30)(18,25,27,21,28,32,20,29);(1,3,6,4)(2,9,11,10)(5,12,8,13)(7,14,16,15)(17,20,22,28,24,27,23,18)(19,29,30,32,26,21,31,25);(1,4,6,3)(2,10,11,9)(5,13,8,12)(7,15,16,14)(17,21,24,29)(18,26,28,19)(20,31,27,30)(22,25,23,32);(3,4)(9,10)(12,13)(14,15)(17,22,24,23)(18,27,28,20)(19,30,26,31)(21,32,29,25)

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,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, 6264910);