#8810312

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

4444 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717508);