#10045515

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

4655 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)(9,10,14,15)(11,16,18,13)(12,17,20,23)(19,21,24,22)(25,26)(27,30)(28,31)(29,34)(32,37)(33,36)(35,39)(38,40);(1,2,4,6)(3,5,8,7)(10,15)(11,18)(17,23)(19,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2,4,6)(3,5,8,7)(12,20)(17,23)(19,24)(21,22)(26,27)(31,33)(32,34)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,21)(20,24)(22,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,2)(3,7)(4,6)(5,8)(9,12)(10,17)(11,19)(13,22)(14,20)(15,23)(16,21)(18,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,3,4,8)(2,5,6,7)(9,13,14,16)(10,11,15,18)(12,21,20,22)(17,24,23,19)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,d^2=i,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, 9952711);