#507803

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 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)(17,18,23,31,19,28,30,25,20,26,21,27,22,24,29,32);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,20,22)(18,24,26,28)(21,29,23,30)(25,31,32,27);(10,13)(14,16)(18,25)(19,22)(21,30)(23,29)(24,27)(26,32)(28,31);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,26)(24,28)(25,32)(27,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=gi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414999);