#10040831

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

4266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=i,e^2=h,f^2=h,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, 9948027);