#8917804

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

4478 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)(9,10,14,16)(11,15,19,24)(12,17,21,23)(13,18,20,22)(25,26,31,30)(27,36,39,32)(28,33,38,34)(29,35,37,40);(3,7)(5,8)(9,11,14,19)(10,15,16,24)(12,20,21,13)(17,22,23,18)(25,27)(26,32)(28,37)(29,38)(30,36)(31,39)(33,35)(34,40);(1,2)(3,5)(4,6)(7,8)(10,16)(11,19)(12,21)(18,22)(25,28,31,38)(26,33,30,34)(27,37,39,29)(32,35,36,40);(9,12)(10,17)(11,20)(13,19)(14,21)(15,22)(16,23)(18,24)(25,28)(26,34)(27,37)(29,39)(30,33)(31,38)(32,40)(35,36);(1,3)(2,5)(4,7)(6,8)(9,13,14,20)(10,18,16,22)(11,21,19,12)(15,23,24,17)(25,29,31,37)(26,35,30,40)(27,38,39,28)(32,33,36,34);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,24)(20,22)(21,23)(25,30,31,26)(27,36,39,32)(28,33,38,34)(29,40,37,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8825000);