#487371

order	512
nilpotency class	4
isoclinism	isoclinic class of #486875
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×C₄⋊C₄	C₂⁴
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1026 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)(3,10)(4,8)(5,9)(6,13)(7,15)(11,21)(12,18)(14,25)(16,29)(17,26)(19,27)(20,32)(22,28)(23,31)(24,30);(2,8)(3,16)(4,21)(5,23)(9,10)(11,13)(12,29)(14,27)(15,28)(17,20)(18,24)(19,22)(26,31)(30,32);(2,9)(3,17)(4,21)(5,18)(8,10)(11,32)(12,15)(13,30)(14,31)(16,20)(19,22)(23,24)(26,27)(28,29);(1,3,6,20)(2,10,13,32)(4,18,21,23)(5,19,24,22)(7,16,25,17)(8,12,11,31)(9,27,30,28)(14,26,15,29)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394567);