#8165189

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

4381 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8072385);