#8914443

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

4362 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=ghi,f^2=gh,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, 8821639);