#8760539

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

4431 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8667735);