#10040425

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

4441 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)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,26,30,32)(27,31,34,29)(28,36,38,33)(35,39,37,40);(1,3,5,4)(2,6,7,8)(10,14)(11,17)(16,19)(18,20)(25,27,30,34)(26,31,32,29)(28,37,38,35)(33,39,36,40);(1,2)(3,6)(4,8)(5,7)(10,14)(11,17)(16,19)(18,20)(26,32)(27,34)(28,38)(39,40);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,28,30,38)(26,33,32,36)(27,35,34,37)(29,39,31,40);(1,4,5,3)(2,6,7,8)(9,11,13,17)(10,15,14,22)(12,20,21,18)(16,24,19,23)(25,27)(26,31)(28,35)(29,32)(30,34)(33,40)(36,39)(37,38);(1,2)(3,6)(4,8)(5,7)(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,29,30,31)(26,34,32,27)(28,39,38,40)(33,37,36,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=i,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, 9947621);