#6469083

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442595
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	#34963
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|c^2=g,e^2=gh,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^b=fi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376279);