#8159161

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

4505 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)(3,7)(4,6)(5,8)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26)(27,34)(28,31)(29,33)(30,36)(32,37)(35,39)(38,40);(1,2,4,6)(3,5,8,7)(10,14)(15,17)(16,20)(22,23)(25,27)(26,30)(28,35)(29,36)(31,38)(32,39)(33,34)(37,40);(1,2)(3,5)(4,6)(7,8)(10,14)(15,17)(16,20)(22,23)(25,28,29,37)(26,31,33,32)(27,35,36,40)(30,38,34,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28)(26,32)(27,35)(29,37)(30,39)(31,33)(34,38)(36,40);(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,26)(27,30)(28,31)(29,33)(32,37)(34,36)(35,38)(39,40);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,27)(26,30)(28,35)(29,36)(31,38)(32,39)(33,34)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066357);