#9781291

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
prime radical	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9688487);