#8425444

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

4700 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^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, 8332640);