#8741944

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

4513 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)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,30,35)(27,33,36,31)(28,32,37,38)(29,34,39,40);(1,3,6,5)(2,7,4,8)(9,11,14,19)(10,15,13,17)(12,18,21,24)(16,22,20,23)(25,27)(26,31)(28,34)(29,32)(30,36)(33,35)(37,40)(38,39);(1,3,6,5)(2,7,4,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,28,30,37)(26,32,35,38)(27,34,36,40)(29,33,39,31);(3,5)(7,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,28)(26,32)(27,34)(29,31)(30,37)(33,39)(35,38)(36,40);(1,4,6,2)(3,8,5,7)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,27)(26,33)(28,34)(29,38)(30,36)(31,35)(32,39)(37,40);(1,5,6,3)(2,7,4,8)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,29,30,39)(26,34,35,40)(27,32,36,38)(28,33,37,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8649140);