#8361247

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

4462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8268443);