#6346738

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	#62819	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3004 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,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.

Relationships

maximal subgroup of	#72268290, #72268296
maximal normal subgroup of	#72268290, #72268296

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253934);