#8917764

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

4675 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)(26,31)(27,35)(28,37)(34,39);(3,7)(5,8)(9,11,14,19)(10,15,16,24)(12,20,21,13)(17,22,23,18)(25,26)(27,32)(28,33)(29,39)(30,31)(34,38)(35,36)(37,40);(1,2)(3,5)(4,6)(7,8)(10,16)(11,19)(12,21)(18,22)(25,27,30,35)(26,32,31,36)(28,38,37,29)(33,34,40,39);(9,12)(10,17)(11,20)(13,19)(14,21)(15,22)(16,23)(18,24)(25,28,30,37)(26,33,31,40)(27,29,35,38)(32,39,36,34);(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)(26,34)(27,28)(30,38)(31,39)(32,40)(33,36)(35,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,24)(20,22)(21,23)(25,26)(27,36)(28,33)(29,34)(30,31)(32,35)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=g,e^2=i,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, 8824960);