#8132478

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	#42974	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8039674);