#8659172

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

4485 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8566368);