#8334321

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

4617 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=i,e^2=g,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, 8241517);