#8843848

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

4477 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:

(2,5)(3,7)(9,10,14,15)(11,13,18,16)(12,20,21,17)(19,24,22,23)(25,26,31,28)(27,34,32,29)(30,36,38,33)(35,37,39,40);(1,2,6,5)(3,8,7,4)(10,15)(12,21)(13,16)(19,22)(27,32)(29,34)(30,38)(33,36);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(17,20)(19,22)(25,27,31,32)(26,29,28,34)(30,35,38,39)(33,37,36,40);(1,2,6,5)(3,4,7,8)(9,11)(10,16)(12,22)(13,15)(14,18)(17,23)(19,21)(20,24)(25,28,31,26)(27,29,32,34)(30,36,38,33)(35,37,39,40);(1,4,6,8)(2,7,5,3)(9,12,14,21)(10,17,15,20)(11,19,18,22)(13,23,16,24)(25,29)(26,32)(27,28)(30,37)(31,34)(33,39)(35,36)(38,40);(1,5,6,2)(3,8,7,4)(9,13,14,16)(10,11,15,18)(12,23,21,24)(17,19,20,22)(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=gh,d^2=i,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, 8751044);