#487355

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

1020 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)(9,16)(12,25)(13,22)(14,19)(15,24)(17,20)(18,27)(21,28)(23,30)(26,31)(29,32);(2,7)(3,12)(4,16)(8,21)(9,10)(11,20)(13,15)(14,29)(17,31)(19,25)(22,23)(24,32)(26,27)(28,30);(3,13)(4,17)(8,22)(12,15)(14,30)(16,31)(19,32)(21,23)(24,25)(28,29);(1,3,5,15)(2,8,10,24)(4,14,16,30)(6,12,18,13)(7,19,9,23)(11,25,27,22)(17,28,31,29)(20,21,26,32)

Polycyclic presentation:

2^9|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, 394551);