#10519386

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

4729 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10426582);