#8599235

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

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

Polycyclic presentation:

2^9|c^2=hi,d^2=gh,f^2=g,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, 8506431);