#8517473

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

4693 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8424669);