#8445884

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

4339 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)(9,10,15,19)(11,20,21,16)(12,17,22,24)(13,14,18,23)(25,26,32,30)(27,36,39,33)(28,40,34,29)(31,35,38,37);(1,3,6,4)(2,7,5,8)(11,21)(12,22)(16,20)(17,24)(25,27)(26,33)(28,37)(29,31)(30,36)(32,39)(34,35)(38,40);(9,11)(10,16)(12,18)(13,22)(14,17)(15,21)(19,20)(23,24)(25,28)(26,29)(27,37)(30,40)(31,33)(32,34)(35,39)(36,38);(2,5)(7,8)(9,12,15,22)(10,17,19,24)(11,18,21,13)(14,20,23,16)(25,29,32,40)(26,34,30,28)(27,38,39,31)(33,37,36,35);(1,4,6,3)(2,8,5,7)(9,13,15,18)(10,14,19,23)(11,22,21,12)(16,17,20,24)(25,30,32,26)(27,36,39,33)(28,40,34,29)(31,37,38,35);(1,5,6,2)(3,8,4,7)(9,14)(10,18)(11,17)(12,16)(13,19)(15,23)(20,22)(21,24)(25,31,32,38)(26,35,30,37)(27,29,39,40)(28,33,34,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=h,e^2=gi,f^2=g,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, 8353080);