#9061448

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

4551 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8968644);