#9684130

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

4166 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=g,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,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, 9591326);