#8399469

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

4630 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)(3,5,7,8)(9,10)(11,16)(12,18)(13,14)(15,17)(19,24)(20,23)(21,22)(26,32)(28,33)(34,39)(36,40);(1,3,4,7)(2,5,6,8)(9,11,15,20)(10,16,17,23)(12,19,22,14)(13,18,24,21)(25,26)(27,33)(28,29)(30,34)(31,32)(35,40)(36,37)(38,39);(1,2)(3,5)(4,6)(7,8)(10,17)(11,20)(14,19)(18,21)(25,27,31,29)(26,33,32,28)(30,37,38,35)(34,36,39,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,18)(11,19)(13,23)(14,20)(15,22)(16,24)(17,21)(25,28)(26,27)(29,32)(30,36)(31,33)(34,35)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,19)(11,21)(12,16)(14,17)(15,24)(18,20)(22,23)(25,29,31,27)(26,33,32,28)(30,37,38,35)(34,40,39,36);(9,14,15,19)(10,13,17,24)(11,12,20,22)(16,18,23,21)(25,30,31,38)(26,34,32,39)(27,35,29,37)(28,36,33,40)

Polycyclic presentation:

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

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, 8306665);