#6357498

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357480
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
prime radical	#60946	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1938 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,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, 6264694);