#8110590

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4270 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)(3,8)(4,6)(5,7)(9,10,15,18)(11,19,20,16)(12,13,23,24)(14,22,21,17)(25,26,30,33)(27,28,31,36)(29,37,38,32)(34,40,39,35);(2,4)(7,8)(11,20)(12,23)(13,24)(16,19)(25,27)(26,28)(29,34)(30,31)(32,35)(33,36)(37,40)(38,39);(1,3,6,5)(2,7,4,8)(9,11,15,20)(10,16,18,19)(12,21,23,14)(13,22,24,17)(25,28)(26,31)(27,33)(29,35)(30,36)(32,39)(34,37)(38,40);(1,2,6,4)(3,7,5,8)(9,12)(10,13)(11,21)(14,20)(15,23)(16,22)(17,19)(18,24);(1,4,6,2)(3,8,5,7)(9,13)(10,12)(11,22)(14,19)(15,24)(16,21)(17,20)(18,23)(25,28)(26,31)(27,33)(29,35)(30,36)(32,39)(34,37)(38,40);(1,5,6,3)(2,8,4,7)(9,14,15,21)(10,17,18,22)(11,12,20,23)(13,19,24,16)(25,29,30,38)(26,32,33,37)(27,34,31,39)(28,35,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017786);