#10519391

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

4536 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)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,26,32,35)(27,36,30,33)(28,31,29,34)(37,38,39,40);(9,11,15,13)(10,16,18,12)(14,19,22,21)(17,23,24,20)(25,27,32,30)(26,33,35,36)(28,39,29,37)(31,38,34,40);(1,3,6,5)(2,7,4,8)(25,28,32,29)(26,34,35,31)(27,37,30,39)(33,38,36,40);(1,3,6,5)(2,8,4,7)(9,12,15,16)(10,11,18,13)(14,20,22,23)(17,19,24,21)(25,29,32,28)(26,31,35,34)(27,37,30,39)(33,38,36,40);(1,4,6,2)(3,8,5,7)(9,13,15,11)(10,16,18,12)(14,19,22,21)(17,20,24,23)(25,30,32,27)(26,33,35,36)(28,39,29,37)(31,40,34,38);(1,5,6,3)(2,8,4,7)(9,14,15,22)(10,17,18,24)(11,19,13,21)(12,20,16,23)(25,31)(26,28)(27,38)(29,35)(30,40)(32,34)(33,39)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=ghi,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, 10426587);