#8105894

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013090);