#8720734

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

4604 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)(3,8)(4,5)(6,7)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24)(25,26,30,31)(27,32,36,35)(28,38,39,33)(29,34,40,37);(1,3,6,5)(2,4,7,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,26,30,31)(27,35,36,32)(28,33,39,38)(29,34,40,37);(1,3,6,5)(2,4,7,8)(12,21)(16,20)(19,22)(23,24)(26,31)(27,36)(33,38)(34,37);(2,7)(3,5)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24)(25,27,30,36)(26,32,31,35)(28,37,39,34)(29,33,40,38);(1,4,6,8)(2,3,7,5)(9,13,14,11)(10,15,17,18)(12,22,21,19)(16,23,20,24)(25,28)(26,33)(27,37)(29,35)(30,39)(31,38)(32,40)(34,36);(1,5,6,3)(2,4,7,8)(9,10)(11,15)(12,20)(13,18)(14,17)(16,21)(19,24)(22,23)(25,29,30,40)(26,34,31,37)(27,38,36,33)(28,32,39,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,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, 8627930);