#8720469

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

4400 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=i,e^2=h,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, 8627665);