#8765598

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

4513 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=gh,e^2=gh,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, 8672794);