#8823859

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

4524 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731055);