#507526

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=g,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414722);