#8936595

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

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

4457 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8843791);