#8165175

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

4460 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)(10,14)(11,17)(16,22)(18,19)(25,26,29,33)(27,31,34,30)(28,36,37,32)(35,38,40,39);(1,2,4,6)(3,5,8,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(27,34)(30,31)(35,40)(38,39);(1,2)(3,5)(4,6)(7,8)(11,17)(15,21)(18,19)(23,24)(25,27,29,34)(26,30,33,31)(28,35,37,40)(32,38,36,39);(1,3,4,8)(2,5,6,7)(9,11,13,17)(10,15,14,21)(12,19,20,18)(16,24,22,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,26)(27,30)(28,32)(29,33)(31,34)(35,38)(36,37)(39,40);(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,28,29,37)(26,32,33,36)(27,35,34,40)(30,38,31,39)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=g,e^2=ghi,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, 8072371);