#8459168

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

4491 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8366364);