#10038549

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

4339 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)(11,16)(12,17)(13,21)(14,15)(18,22)(19,23)(20,24)(25,26)(27,30)(28,36)(29,31)(32,34)(33,37)(35,39)(38,40);(1,3,5,4)(2,6,7,8)(10,15)(11,18)(12,21)(20,23)(25,27,29,34)(26,30,31,32)(28,35,37,40)(33,38,36,39);(1,2)(3,6)(4,8)(5,7)(10,15)(11,18)(12,21)(20,23)(26,31)(27,34)(33,36)(35,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,18)(10,16,15,22)(12,19,21,24)(13,20,17,23)(25,27)(26,32)(28,35)(29,34)(30,31)(33,39)(36,38)(37,40);(1,4,5,3)(2,6,7,8)(9,12,14,21)(10,17,15,13)(11,19,18,24)(16,23,22,20)(25,26)(27,32)(28,36)(29,31)(30,34)(33,37)(35,38)(39,40);(1,2)(3,6)(4,8)(5,7)(9,13,14,17)(10,12,15,21)(11,20,18,23)(16,19,22,24)(25,28,29,37)(26,33,31,36)(27,35,34,40)(30,38,32,39)

Polycyclic presentation:

2^9|b^2=gh,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, 9945745);