#8158964

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

4289 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=g,f^2=gh,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, 8066160);