#8139497

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

4401 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,e^2=g,f^2=gh,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, 8046693);