#483815

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	C₂×#508	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=i,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	#72216701, #72216702, #72216898, …
maximal normal subgroup of	#72216701, #72216702, #72216898, …
maximal quotient of	C₂×#483815, #72318512, #72318600, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391011);