#8371246

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4201 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)(3,5)(4,6)(7,8)(9,10,15,12)(11,16,18,14)(13,17,23,20)(19,24,21,22)(25,26,32,35)(27,36,31,33)(28,30,29,34)(37,39,38,40);(1,2,4,6)(3,5,7,8)(9,11,15,18)(10,16,12,14)(13,21,23,19)(17,22,20,24)(25,27,32,31)(26,33,35,36)(28,38,29,37)(30,39,34,40);(3,7)(5,8)(11,18)(14,16)(19,21)(22,24)(25,28,32,29)(26,34,35,30)(27,37,31,38)(33,39,36,40);(1,2,4,6)(3,5,7,8)(9,12,15,10)(11,14,18,16)(13,17,23,20)(19,24,21,22)(25,29,32,28)(26,30,35,34)(27,38,31,37)(33,40,36,39);(1,2)(3,5)(4,6)(7,8)(9,13,15,23)(10,17,12,20)(11,19,18,21)(14,22,16,24)(25,30)(26,28)(27,39)(29,35)(31,40)(32,34)(33,38)(36,37);(1,3,4,7)(2,5,6,8)(9,14)(10,11)(12,18)(13,22)(15,16)(17,19)(20,21)(23,24)(25,31,32,27)(26,33,35,36)(28,38,29,37)(30,40,34,39)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=ghi,e^2=hi,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, 8278442);