#8678978

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	C₂×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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)(11,19)(13,23)(16,20)(18,24)(25,26,31,34)(27,32,30,33)(28,37,38,29)(35,40,39,36);(2,5)(6,7)(9,10,15,14)(11,20,19,16)(12,22,21,17)(13,18,23,24)(25,26)(27,32)(28,29)(30,33)(31,34)(35,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,19)(10,16,14,20)(12,13,21,23)(17,18,22,24)(25,27,31,30)(26,32,34,33)(28,35,38,39)(29,36,37,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,21)(10,17,14,22)(11,13,19,23)(16,18,20,24)(25,28)(26,29)(27,35)(30,39)(31,38)(32,36)(33,40)(34,37);(9,13)(10,18)(11,21)(12,19)(14,24)(15,23)(16,22)(17,20)(25,29)(26,28)(27,36)(30,40)(31,37)(32,35)(33,39)(34,38);(9,14,15,10)(11,20,19,16)(12,17,21,22)(13,18,23,24)(25,30,31,27)(26,33,34,32)(28,35,38,39)(29,36,37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8586174);