#8750116

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

4239 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=g,e^2=gh,f^2=gi,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, 8657312);