#8362059

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

4265 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:

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

Polycyclic presentation:

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

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, 8269255);