#486575

order	512
nilpotency class	4
isoclinism	isoclinic class of #486139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#2535	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278345, #72278365
maximal normal subgroup of	#72278345, #72278365
maximal quotient of	#72278432

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393771);