#8722264

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

4563 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,e^2=g,f^2=gi,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, 8629460);