#496009

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237289, #72239540, #72239543, …
maximal normal subgroup of	#72237289, #72239540, #72239543, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403205);