#6357689

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

2013 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)(4,10)(6,12)(8,13)(11,15)(14,16)(17,18,23,19)(20,30,31,32)(21,25,27,28)(22,29,24,26);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,22,29,23,18,24,26)(20,27,28,32,31,21,25,30);(1,3,10,15,5,11,4,9)(2,6,13,16,7,14,8,12)(17,20,22,28,23,31,24,25)(18,21,26,30,19,27,29,32);(2,7)(6,14)(8,13)(12,16)(17,21)(18,25)(19,28)(20,26)(22,30)(23,27)(24,32)(29,31);(20,31)(21,27)(25,28)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=ghi,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, 6264885);