#6347256

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347252
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 ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3289 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)(3,11)(4,14)(8,20)(9,15)(10,18)(12,26)(13,25)(16,29)(19,22)(21,31)(23,30)(24,28)(27,32);(1,2)(3,8)(4,15)(5,9)(6,10)(7,14)(11,22)(12,27)(13,21)(16,30)(17,23)(18,29)(19,28)(20,26)(24,31)(25,32);(4,16)(7,18)(12,24)(14,29)(15,30)(19,20)(26,28)(27,31);(1,3,5,13)(2,8,9,21)(4,12,14,28)(6,11,17,25)(7,19,15,27)(10,22,23,32)(16,24,29,26)(18,20,30,31);(2,9)(7,15)(8,21)(10,23)(18,30)(19,27)(20,31)(22,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213216, #72213613, #72213821, …
maximal normal subgroup of	#72213216, #72213613, #72213821, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254452);