#8843920

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

4583 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)(9,10,15,18)(11,16,12,17)(13,23,24,14)(19,20,21,22)(25,26,32,36)(27,34,29,33)(28,30,31,35)(37,40,39,38);(1,2,4,6)(3,7,8,5)(9,11,15,12)(10,16,18,17)(13,19,24,21)(14,22,23,20)(25,27,32,29)(26,33,36,34)(28,37,31,39)(30,38,35,40);(1,3)(2,5)(4,8)(6,7)(11,12)(16,17)(19,21)(20,22)(25,28,32,31)(26,30,36,35)(27,37,29,39)(33,38,34,40);(3,8)(5,7)(9,12,15,11)(10,17,18,16)(13,19,24,21)(14,22,23,20)(25,29,32,27)(26,34,36,33)(28,37,31,39)(30,38,35,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,14)(11,19)(12,21)(15,24)(16,22)(17,20)(18,23)(25,30,32,35)(26,28,36,31)(27,38,29,40)(33,37,34,39);(1,2)(3,5)(4,6)(7,8)(9,14)(10,13)(11,20)(12,22)(15,23)(16,21)(17,19)(18,24)(25,31,32,28)(26,35,36,30)(27,37,29,39)(33,38,34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8751116);