#9054422

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

4191 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)(3,7)(4,5)(6,8)(9,10,15,17)(11,16,12,21)(13,18,23,22)(14,20,24,19)(25,26,31,28)(27,34,33,30)(29,32,37,36)(35,40,39,38);(2,5)(6,7)(9,11,15,12)(10,16,17,21)(13,20,23,19)(14,18,24,22)(29,37)(32,36)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(10,17)(11,12)(13,23)(14,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,40)(37,39);(1,3,4,8)(2,6,5,7)(9,12,15,11)(10,16,17,21)(13,20,23,19)(14,22,24,18)(25,28,31,26)(27,34,33,30)(29,36,37,32)(35,40,39,38);(9,13)(10,18)(11,20)(12,19)(14,21)(15,23)(16,24)(17,22)(25,29,31,37)(26,32,28,36)(27,35,33,39)(30,38,34,40);(9,14,15,24)(10,19,17,20)(11,18,12,22)(13,21,23,16)(25,30,31,34)(26,33,28,27)(29,38,37,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961618);