#6529319

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500010
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3285 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=hi,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,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	#72268399, #72268427
maximal normal subgroup of	#72268399, #72268427

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6436515);