#9778471

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	#44082	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4208 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)(26,30)(27,33)(32,35)(34,36);(2,5)(6,7)(9,10,15,18)(11,12,13,16)(14,17,22,24)(19,20,21,23)(25,26,29,30)(27,31,33,38)(28,35,37,32)(34,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,18,12)(14,19,22,21)(17,23,24,20)(25,27)(26,31)(28,34)(29,33)(30,38)(32,39)(35,40)(36,37);(1,3)(2,6)(4,8)(5,7)(9,12,15,16)(10,11,18,13)(14,20,22,23)(17,19,24,21)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40);(9,13,15,11)(10,16,18,12)(14,19,22,21)(17,20,24,23)(25,28,29,37)(26,32,30,35)(27,34,33,36)(31,39,38,40);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,27)(26,31)(28,36)(29,33)(30,38)(32,40)(34,37)(35,39)

Polycyclic presentation:

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

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, 9685667);