#8270480

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

4444 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=hi,e^2=gi,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, 8177676);