#9390451

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	#48830	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9297647);