#8703519

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

4301 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=gh,e^2=h,f^2=g,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, 8610715);