#8106046

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

4504 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)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,31,33)(27,36,38,32)(28,39,35,30)(29,40,37,34);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27,31,38)(26,32,33,36)(28,40,35,34)(29,30,37,39);(2,5)(6,8)(12,21)(16,20)(18,24)(22,23)(26,33)(28,35)(32,36)(34,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,28,31,35)(26,30,33,39)(27,34,38,40)(29,36,37,32);(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,29,31,37)(26,34,33,40)(27,30,38,39)(28,36,35,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,30)(26,35)(27,37)(28,33)(29,38)(31,39)(32,34)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,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, 8013242);