#9065820

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

4522 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)(11,12)(13,21)(14,15)(16,23)(17,18)(19,22)(20,24)(25,26,31,29)(27,33,28,32)(30,37,38,34)(35,39,36,40);(3,7)(5,8)(9,11)(10,12)(13,19)(14,17)(15,18)(16,20)(21,22)(23,24)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(1,2)(3,5)(4,6)(7,8)(10,15)(11,17)(16,21)(19,24)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(9,12)(10,11)(13,22)(14,18)(15,17)(16,24)(19,21)(20,23)(25,28,31,27)(26,33,29,32)(30,35,38,36)(34,39,37,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,18)(12,17)(13,16)(14,15)(19,22)(20,24)(21,23)(25,29,31,26)(27,32,28,33)(30,37,38,34)(35,39,36,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,19)(12,20)(14,23)(15,21)(17,24)(18,22)(25,30,31,38)(26,34,29,37)(27,35,28,36)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=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, 8973016);