#487364

order	512
nilpotency class	4
isoclinism	isoclinic class of #486867
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×C₄⋊C₄	C₂⁴
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

870 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)(19,23)(20,22)(21,25)(24,28)(26,30)(27,29)(31,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(19,26)(20,28)(23,30)(29,31);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)(17,19,21,26)(18,23,25,30)(20,27,28,32)(22,29,24,31)

Polycyclic presentation:

2^9|d^2=gi,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=dg,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, 394560);