#487361

order	512
nilpotency class	4
isoclinism	isoclinic class of #486811
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 ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

905 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)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,31)(30,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(17,19,22,20)(18,23,26,31)(21,28,30,29)(24,25,27,32);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(17,20,22,19)(18,24,26,27)(21,28,30,29)(23,25,31,32);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)(19,20)(23,31)(24,27)(28,29)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394557);