#6354847

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354835
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
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1643 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^c=eh,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, 6262043);