#6347328

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	#62788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2868 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=hi,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	#72271341, #72271566, #72271570
maximal normal subgroup of	#72271341, #72271566, #72271570

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254524);