#8718423

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

4485 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,22)(13,18)(14,15)(16,20)(17,23)(21,24)(25,26,30,34)(27,29,33,31)(28,36,37,32)(35,38,40,39);(1,2,4,6)(3,5,8,7)(10,15)(11,20)(17,22)(18,21)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(10,15)(11,20)(17,22)(18,21)(28,37)(32,36)(35,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,24)(18,23)(25,28,30,37)(26,32,34,36)(27,35,33,40)(29,39,31,38);(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,16,24,19)(25,27)(26,29)(28,35)(30,33)(31,34)(32,39)(36,38)(37,40);(9,13,14,24)(10,18,15,21)(11,17,20,22)(12,19,23,16)(25,29)(26,33)(27,34)(28,38)(30,31)(32,35)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625619);