#8759304

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

4611 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)(11,16)(12,18)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26,30,34)(27,28,32,31)(29,33,38,40)(35,36,39,37);(1,2)(3,7)(4,5)(6,8)(9,11,15,12)(10,16,17,18)(13,22,19,20)(14,23,24,21)(25,27,30,32)(26,31,34,28)(29,35,38,39)(33,37,40,36);(2,5)(6,7)(10,17)(11,12)(13,19)(21,23)(25,28,30,31)(26,32,34,27)(29,37,38,36)(33,35,40,39);(1,2)(3,6)(4,5)(7,8)(9,12,15,11)(10,18,17,16)(13,22,19,20)(14,23,24,21)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,19)(10,14,17,24)(11,20,12,22)(16,21,18,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40);(1,3)(2,6)(4,8)(5,7)(9,14)(10,19)(11,21)(12,23)(13,17)(15,24)(16,22)(18,20)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,e^2=h,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, 8666500);