#8614116

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
prime radical	#51687	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4434 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8521312);