#8425643

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

4411 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332839);