#9498320

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

4231 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9405516);