#486095

order	512
nilpotency class	4
isoclinism	isoclinic class of #486091
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
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

966 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18)(19,23)(20,22)(21,24)(25,31)(26,29)(27,32)(28,30);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(18,22)(23,29)(24,30)(31,32);(17,19,21,25)(18,23,24,31)(20,27,28,26)(22,29,30,32);(2,7)(3,10)(8,12)(11,13)(19,25)(23,31)(26,27)(29,32)

Polycyclic presentation:

2^9|c^2=i,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,d^c=di,e^c=eh,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284163, #72284218, #72284282
maximal normal subgroup of	#72284163, #72284218, #72284282

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393291);