#6489750

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470225
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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285101, #72285186, #72285362, …
maximal normal subgroup of	#72285101, #72285186, #72285362, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396946);