#8256590

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

4374 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)(3,4)(5,6)(7,8)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,39,37,36);(2,6)(3,8)(10,12)(15,17)(16,20)(22,23)(25,27,30,35)(26,31,33,28)(29,34,38,37)(32,39,40,36);(2,6)(4,7)(13,21)(16,20)(18,24)(22,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,38)(39,40);(1,3)(2,7)(4,6)(5,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23);(1,3)(2,7)(4,6)(5,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(1,4,5,7)(2,3,6,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,38)(26,32,33,40)(27,34,35,37)(28,36,31,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8163786);