#8159034

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

4499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066230);