#8902467

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

4444 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8809663);