#8256626

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

4291 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:

(2,4)(6,7)(9,10,15,14)(11,18,16,12)(13,21,22,17)(19,20,23,24)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,39,37,36);(1,2,5,4)(3,7,8,6)(9,11,15,16)(10,12,14,18)(13,19,22,23)(17,20,21,24)(25,27,30,35)(26,31,33,28)(29,34,38,37)(32,39,40,36);(1,3,5,8)(2,6,4,7)(9,12,15,18)(10,16,14,11)(13,20,22,24)(17,23,21,19)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,38)(39,40);(1,4,5,2)(3,7,8,6)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,24)(22,23);(9,13,15,22)(10,17,14,21)(11,19,16,23)(12,20,18,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(9,14,15,10)(11,18,16,12)(13,21,22,17)(19,24,23,20)(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,c^2=g,d^2=gh,e^2=ghi,f^2=ghi,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, 8163822);