#8358067

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

4704 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=i,e^2=g,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, 8265263);