#6357549

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

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:

(1,2,7,11)(3,12,14,15)(4,9,8,16)(5,10,6,13)(17,18,23,28)(19,29,31,25)(20,32,30,24)(21,26,22,27);(1,3,6,9,7,14,5,16)(2,8,10,12,11,4,13,15)(17,19,21,30,23,31,22,20)(18,24,27,29,28,32,26,25);(1,4)(2,9)(3,13)(5,12)(6,15)(7,8)(10,14)(11,16)(17,20,22,31,23,30,21,19)(18,25,26,32,28,29,27,24);(1,5,7,6)(2,10,11,13)(3,9,14,16)(4,12,8,15)(17,21,23,22)(18,26,28,27)(19,30,31,20)(24,25,32,29);(3,14)(4,8)(9,16)(12,15)(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=i,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, 6264745);