#483811

order	512
nilpotency class	4
isoclinism	isoclinic class of #483807
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	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

472 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320804, #72320825, #72320847, …
maximal normal subgroup of	#72320804, #72320825, #72320847, …
maximal quotient of	#72343621

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391007);