#8720790

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

4429 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)(3,6,8,4)(9,10,15,17)(11,19,13,16)(12,14,20,18)(21,22,23,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,11,15,13)(10,16,17,19)(12,23,20,21)(14,22,18,24)(26,27)(31,33)(32,34)(38,39);(1,2)(3,4)(5,7)(6,8)(10,17)(12,20)(16,19)(21,23)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,3,5,8)(2,6,7,4)(9,12)(10,18)(11,21)(13,23)(14,17)(15,20)(16,22)(19,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,2)(3,6)(4,8)(5,7)(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,22,18,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4,5,6)(2,3,7,8)(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,21,19,23)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627986);