#487356

order	512
nilpotency class	4
isoclinism	isoclinic class of #486811
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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 ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1067 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341031, #72341196, #72341202
maximal normal subgroup of	#72341031, #72341202

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394552);