#8843828

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

4483 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)(3,5)(4,7)(6,8)(9,10,15,18)(11,16,12,17)(13,23,24,14)(19,20,21,22)(25,26,31,28)(27,34,32,29)(30,36,38,33)(35,37,39,40);(1,2,6,8)(3,5,7,4)(9,11,15,12)(10,16,18,17)(13,19,24,21)(14,22,23,20)(27,32)(29,34)(30,38)(33,36);(1,3,6,7)(2,4,8,5)(11,12)(16,17)(19,21)(20,22)(25,27,31,32)(26,29,28,34)(30,35,38,39)(33,37,36,40);(1,4,6,5)(2,3,8,7)(9,12,15,11)(10,17,18,16)(13,19,24,21)(14,22,23,20)(25,28,31,26)(27,29,32,34)(30,36,38,33)(35,37,39,40);(1,2)(3,4)(5,7)(6,8)(9,13)(10,14)(11,19)(12,21)(15,24)(16,22)(17,20)(18,23)(25,29)(26,32)(27,28)(30,37)(31,34)(33,39)(35,36)(38,40);(1,5,6,4)(2,7,8,3)(9,14)(10,13)(11,20)(12,22)(15,23)(16,21)(17,19)(18,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=i,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, 8751024);