#8108771

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

4574 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=gh,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, 8015967);