#10036773

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

4518 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,31,28)(27,34,30,32)(29,33,37,36)(35,40,38,39);(1,2)(3,8)(4,7)(5,6)(11,17)(15,21)(18,23)(22,24)(25,27,31,30)(26,32,28,34)(29,35,37,38)(33,39,36,40);(2,6)(3,7)(10,14)(15,21)(16,19)(22,24)(29,37)(33,36)(35,38)(39,40);(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,31,26)(27,32,30,34)(29,36,37,33)(35,39,38,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29,31,37)(26,33,28,36)(27,35,30,38)(32,39,34,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,27)(26,32,28,34)(29,35,37,38)(33,40,36,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9943969);