#8188139

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

4476 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)(9,10,15,17)(11,18,13,16)(12,21,23,14)(19,20,22,24)(25,26,31,36)(27,35,39,32)(28,33,29,38)(30,34,40,37);(1,2)(3,5)(4,6)(7,8)(9,11,15,13)(10,16,17,18)(12,19,23,22)(14,20,21,24)(25,27)(26,32)(28,37)(29,34)(30,38)(31,39)(33,40)(35,36);(1,2,4,6)(3,5,8,7)(10,17)(14,21)(16,18)(20,24)(25,28,31,29)(26,33,36,38)(27,37,39,34)(30,32,40,35);(9,12)(10,14)(11,19)(13,22)(15,23)(16,20)(17,21)(18,24)(25,29,31,28)(26,33,36,38)(27,37,39,34)(30,35,40,32);(1,3)(2,5)(4,8)(6,7)(9,13,15,11)(10,18,17,16)(12,22,23,19)(14,24,21,20)(25,30,31,40)(26,34,36,37)(27,38,39,33)(28,32,29,35);(1,2)(3,5)(4,6)(7,8)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,23)(18,22)(25,27)(26,35)(28,34)(29,37)(30,38)(31,39)(32,36)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095335);