#8348623

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

4462 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=g,e^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, 8255819);