#8935343

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

4528 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)(11,14,17,18)(12,15,20,22)(19,21,23,24)(25,26,31,29)(27,33,28,32)(30,37,38,34)(35,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(2,5)(3,7)(11,17)(12,20)(14,18)(15,22)(26,29)(30,38)(32,33)(35,36);(3,7)(6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,28,31,27)(26,33,29,32)(30,36,38,35)(34,40,37,39);(1,3)(2,6)(4,7)(5,8)(9,11,13,17)(10,14,16,18)(12,19,20,23)(15,21,22,24)(25,29,31,26)(27,32,28,33)(30,37,38,34)(35,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,12,13,20)(10,15,16,22)(11,19,17,23)(14,21,18,24)(25,30,31,38)(26,34,29,37)(27,35,28,36)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=g,f^2=g,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, 8842539);