#10040707

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

4336 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)(3,8)(4,6)(5,7)(9,10,15,17)(11,16,14,24)(12,20,22,13)(18,23,21,19)(25,26)(27,29)(28,34)(30,33)(31,32)(35,38)(36,39)(37,40);(1,3,5,4)(2,6,7,8)(9,11,15,14)(10,16,17,24)(12,21,22,18)(13,23,20,19)(26,32)(28,36)(29,33)(35,37);(1,2)(3,6)(4,8)(5,7)(9,12,15,22)(10,13,17,20)(11,18,14,21)(16,19,24,23)(25,27,31,30)(26,33,32,29)(28,37,36,35)(34,38,39,40);(1,2)(3,8)(4,6)(5,7)(9,12)(10,13)(11,18)(14,21)(15,22)(16,19)(17,20)(23,24)(25,28,31,36)(26,34,32,39)(27,35,30,37)(29,38,33,40);(1,4,5,3)(2,6,7,8)(9,13)(10,12)(11,19)(14,23)(15,20)(16,18)(17,22)(21,24)(25,29,31,33)(26,27,32,30)(28,38,36,40)(34,35,39,37);(1,2)(3,6)(4,8)(5,7)(9,14,15,11)(10,16,17,24)(12,21,22,18)(13,19,20,23)(25,30,31,27)(26,29,32,33)(28,37,36,35)(34,38,39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=hi,e^2=g,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, 9947903);