#8697516

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

4629 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8604712);