#8726451

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4232 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=gh,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633647);