#8750721

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

4195 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657917);