#6346708

order	512
nilpotency class	3
isoclinism	isoclinic class of #6346696
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
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232880, #72234669, #72234686
maximal normal subgroup of	#72232880, #72234669, #72234686

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253904);