#6355451

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355448
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1752 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,13)(15,16)(18,23)(19,27)(20,21)(25,26)(28,29)(30,32);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,15)(12,16)(17,18,20,25,22,26,21,23)(19,24,27,32,28,31,29,30);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(3,10)(6,14)(11,16)(13,15)(19,28)(24,31)(27,29)(30,32);(2,8)(6,14)(7,9)(13,15)(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,32,31,30)

Polycyclic presentation:

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

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, 6262647);