#8436548

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

4414 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)(3,4,7,8)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,22,20,24)(25,26,32,30)(27,36,38,33)(28,39,34,29)(31,35,40,37);(1,3,6,7)(2,4,5,8)(10,16)(11,12)(18,21)(19,20)(25,27,32,38)(26,33,30,36)(28,31,34,40)(29,35,39,37);(1,4)(2,3)(5,7)(6,8)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,22,21,24)(25,28)(26,29)(27,31)(30,39)(32,34)(33,35)(36,37)(38,40);(1,5,6,2)(3,4,7,8)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22)(25,29,32,39)(26,34,30,28)(27,37,38,35)(31,36,40,33);(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24)(25,30,32,26)(27,36,38,33)(28,39,34,29)(31,37,40,35);(1,4)(2,7)(3,5)(6,8)(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,19,21,20)(25,31,32,40)(26,35,30,37)(27,34,38,28)(29,33,39,36)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,d^2=h,e^2=g,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, 8343744);