#8260802

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

4491 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)(3,5,7,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,22)(18,23)(21,24)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(1,3,4,7)(2,5,6,8)(9,11,13,19)(10,14,15,17)(12,21,22,18)(16,24,20,23)(26,32)(27,28)(34,37)(35,36);(1,2)(3,5)(4,6)(7,8)(10,15)(14,17)(16,20)(23,24)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(1,3)(2,5)(4,7)(6,8)(9,12,13,22)(10,16,15,20)(11,18,19,21)(14,23,17,24)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,24)(21,22)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=i,d^2=g,e^2=i,f^2=i,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, 8167998);