#6357527

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357516
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 ◁ C₂²×C₄² ◁ 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^b=eh,e^c=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, 6264723);