#8159267

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

4328 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)(3,5)(4,7)(6,8)(9,10,14,17)(11,18,15,12)(13,21,22,16)(19,20,23,24)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(1,3)(2,4)(5,8)(6,7)(9,11,14,15)(10,12,17,18)(13,19,22,23)(16,20,21,24)(26,31)(27,34)(33,36)(35,40);(1,4,6,5)(2,7,8,3)(9,12)(10,15)(11,17)(13,20)(14,18)(16,23)(19,21)(22,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(2,8)(4,5)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,5,6,4)(2,3,8,7)(9,12)(10,15)(11,17)(13,20)(14,18)(16,23)(19,21)(22,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,3)(2,4)(5,8)(6,7)(9,13,14,22)(10,16,17,21)(11,19,15,23)(12,20,18,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,e^2=gh,f^2=h,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, 8066463);