#8695318

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

4421 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8602514);