#8463273

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
prime radical	#42887	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4340 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=gh,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370469);