#10040641

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=ghi,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, 9947837);