#496168

order	512
nilpotency class	4
isoclinism	isoclinic class of #483962
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(11,19,21,33)(12,23)(13,25)(15,26,28,38)(17,31)(18,30)(20,32,22,36)(24,35)(27,37,34,39)(29,40);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,26)(20,34)(21,28)(22,27)(23,31)(24,29)(25,30)(32,39)(33,38)(35,40)(36,37);(9,11,12,19,13,22,23,32,14,21,24,33,25,20,35,36)(10,15,17,26,18,27,31,39,16,28,29,38,30,34,40,37);(1,4,5,3)(2,6,7,8)(10,16)(11,20,21,22)(12,24)(15,27,28,34)(18,30)(19,32,33,36)(23,35)(26,37,38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=eg,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313605, #72313619, #72313647, …
maximal normal subgroup of	#72313605, #72313619, #72313647, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403364);