#8132482

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

4425 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=gi,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^c=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, 8039678);