#6357680

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

1913 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gh,e^2=h,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, 6264876);