#6357532

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^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, 6264728);