#8760923

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

4325 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=h,f^2=gh,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, 8668119);