#8348246

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

4563 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)(11,16)(12,18)(13,21)(14,20)(15,17)(19,24)(22,23)(25,26,31,34)(27,29,28,32)(30,33,38,40)(35,37,36,39);(1,2)(3,7)(4,5)(6,8)(9,11,15,14)(10,16,17,20)(12,21,22,19)(13,23,24,18)(25,27,31,28)(26,32,34,29)(30,35,38,36)(33,39,40,37);(2,5)(6,7)(10,17)(11,14)(18,23)(19,21)(25,27,31,28)(26,29,34,32)(30,36,38,35)(33,39,40,37);(3,8)(6,7)(9,12,15,22)(10,18,17,23)(11,19,14,21)(13,16,24,20)(25,28,31,27)(26,29,34,32)(30,36,38,35)(33,37,40,39);(1,3)(2,6)(4,8)(5,7)(9,13)(10,19)(11,18)(12,20)(14,23)(15,24)(16,22)(17,21)(25,29)(26,28)(27,34)(30,37)(31,32)(33,36)(35,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,14,15,11)(10,20,17,16)(12,21,22,19)(13,23,24,18)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gh,f^2=h,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, 8255442);