#8267603

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

4466 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)(6,8)(26,27)(28,35)(32,34)(36,40);(3,7)(6,8)(9,10)(11,14)(12,16)(13,21)(15,18)(17,23)(19,24)(20,22)(25,26,30,27)(28,33,35,31)(29,32,37,34)(36,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,18,14)(13,19,23,20)(17,24,21,22)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,16,18,14)(13,20,23,19)(17,24,21,22)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,23)(10,17,18,21)(11,19,12,20)(14,22,16,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(9,14,15,16)(10,11,18,12)(13,22,23,24)(17,19,21,20)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=ghi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8174799);