#8709160

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

4517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,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, 8616356);