#8898935

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

4666 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)(3,4)(5,6)(7,8)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26,31,33)(27,32,37,38)(28,34,36,30)(29,35,39,40);(2,6)(4,7)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,27)(26,32)(28,35)(29,30)(31,37)(33,38)(34,39)(36,40);(1,2,5,6)(3,7,8,4)(10,15)(14,17)(16,23)(22,24)(26,33)(27,37)(30,34)(35,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,28,31,36)(26,34,33,30)(27,35,37,40)(29,32,39,38);(1,4,5,7)(2,3,6,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29)(26,35)(27,34)(28,38)(30,37)(31,39)(32,36)(33,40);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,30,31,34)(26,36,33,28)(27,29,37,39)(32,40,38,35)

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806131);