#498540

order	512
nilpotency class	4
isoclinism	isoclinic class of #484139
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 ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

446 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)(3,6)(4,5)(7,8)(9,10)(11,20,24,35)(12,17)(13,26)(14,29)(15,19)(16,23,31,38)(18,28)(21,33,22,34)(25,32)(27,39)(30,37,40,36);(2,5)(3,7)(11,21,24,22)(13,27)(16,30,31,40)(18,32)(20,33,35,34)(23,37,38,36)(25,28)(26,39);(9,11,13,23,14,21,28,36,15,24,27,38,12,22,25,37)(10,16,18,20,17,30,26,34,19,31,32,35,29,40,39,33);(1,3,4,7)(2,6,5,8)(9,12,15,14)(10,17,19,29)(11,22,24,21)(13,25,27,28)(16,30,31,40)(18,26,32,39)(20,34,35,33)(23,37,38,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309671, #72310187, #72310206, …
maximal normal subgroup of	#72309671, #72310187, #72310206, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405736);