#8346698

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

4505 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(27,35)(29,39)(31,36)(33,40);(1,2)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26)(27,36)(28,38)(29,40)(30,34)(31,35)(32,37)(33,39);(2,5)(6,7)(12,20)(15,22)(18,24)(21,23)(25,27,30,35)(26,31,34,36)(28,29,37,39)(32,40,38,33);(3,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28)(26,32)(27,29)(30,37)(31,40)(33,36)(34,38)(35,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,29)(26,33)(27,37)(28,35)(30,39)(31,32)(34,40)(36,38);(1,2)(3,6)(4,5)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,38)(39,40)

Polycyclic presentation:

2^9|b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8253894);