#8118680

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

4461 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:

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

Polycyclic presentation:

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

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, 8025876);