#8389349

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

4659 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8296545);