#8334313

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

4641 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8241509);