#8106006

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

4629 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013202);