#8943864

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

4673 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8851060);