#8137019

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

4498 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)(3,7)(4,6)(5,8)(10,11)(13,21)(15,17)(19,24)(25,26,32,36)(27,29,30,33)(28,37,40,34)(31,39,38,35);(1,2,4,6)(3,5,8,7)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,32,30)(26,33,36,29)(28,35,40,39)(31,37,38,34);(1,2)(3,5)(4,6)(7,8)(25,28)(26,34)(27,35)(29,38)(30,39)(31,33)(32,40)(36,37);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,29,32,33)(26,27,36,30)(28,38,40,31)(34,35,37,39);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,30,32,27)(26,29,36,33)(28,39,40,35)(31,34,38,37);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,31,32,38)(26,35,36,39)(27,37,30,34)(28,33,40,29)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044215);