#8599549

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

4723 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506745);