#8132850

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

4457 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=h,f^2=ghi,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, 8040046);