#8351631

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

4422 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=g,e^2=gh,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, 8258827);