#9064639

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	#41066	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4343 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8971835);