#6346735

order	512
nilpotency class	3
isoclinism	isoclinic class of #6346727
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
prime radical	#62818	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3248 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)(9,13)(10,12)(15,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28);(1,2)(3,8)(4,7)(5,13)(6,9)(10,15)(11,16)(12,14)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(1,3)(2,8)(4,12)(5,10)(6,11)(7,14)(9,16)(13,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(1,4,6,5)(2,7,9,13)(3,10,11,12)(8,15,16,14);(2,9)(7,13)(8,16)(14,15)

Polycyclic presentation:

2^9|a^2=i,d^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,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, 6253931);