#8648916

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

4409 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8556112);