#6347324

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347305
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	#62814	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2895 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(2,7)(6,15)(8,12)(9,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26);(3,10)(6,15)(9,16)(11,13)(19,28)(23,32)(26,31)(27,30)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278872, #72278907, #72279011
maximal normal subgroup of	#72278872, #72278907, #72279011

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254520);