#8348309

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

4584 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,e^2=g,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, 8255505);