#487653

order	512
nilpotency class	4
isoclinism	isoclinic class of #487651
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

1033 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)(5,9)(6,11)(10,15)(12,24)(13,23)(14,19)(16,21)(17,20)(18,25)(22,28)(26,30)(27,32)(29,31);(2,7)(3,12)(4,15)(8,21)(9,10)(11,20)(13,27)(14,29)(16,28)(17,30)(19,24)(22,32)(23,31)(25,26);(1,3,6,12)(2,8,11,24)(4,16,17,14)(5,13,18,27)(7,19,20,21)(9,23,25,32)(10,22,26,31)(15,29,30,28);(2,9)(3,13)(7,10)(11,25)(12,27)(14,28)(16,29)(20,26)

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,c^b=ci,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394849);