#6348015

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347957
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=hi,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,f^b=fh,g^b=gi,e^c=eh,f^c=fi

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, 6255211);