#8108742

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
prime radical	#59139	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015938);