#8759153

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

4427 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=gh,f^2=g,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, 8666349);